第 22199 題：有限因數格上的Möbius反演

原題與已知條件

positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={"n":24,"a":4,"b":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={"n":24,"a":4,"b":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-mobius-inversion-finite-divisors

逐步推導

source operation=mobinv-roundtrip-table;original snapshot and mathematics retained

explicit ordered source output fields=[n, roundtrip_rows, all_equal, boundary]

Mobius inversion ORIGINAL roundtrip:actual positive input=24;check complete integer range1..24

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=1 forward:actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=1 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=1 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=1 forward full forward divisor sum := sum(15)

Mobius roundtrip m=1 forward full forward divisor sum = 15

Mobius roundtrip m=1 inverse:actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 := div(1, 1)

Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 = 1

Mobius roundtrip m=1 inverse F(n/1):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum := sum(15)

Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum = 15

Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) := mul(1, 15)

Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) = 15

Mobius roundtrip m=1 inverse complete inverse-term sum := sum(15)

Mobius roundtrip m=1 inverse complete inverse-term sum = 15

Mobius roundtrip m=1 inverse direct ORIGINAL a times n := mul(4, 1)

Mobius roundtrip m=1 inverse direct ORIGINAL a times n = 4

Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=1 inverse direct plus ORIGINAL b = 15

Mobius roundtrip m=1 inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, 15, 15]]

complete reconstructed=15;direct f(n)=15;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=2 forward:actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=2 forward full forward divisor sum := sum(15, 19)

Mobius roundtrip m=2 forward full forward divisor sum = 34

Mobius roundtrip m=2 inverse:actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=2 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=2 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 := div(2, 1)

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 = 2

Mobius roundtrip m=2 inverse F(n/1):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum = 34

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) := mul(1, 34)

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) = 34

Mobius roundtrip m=2 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=2 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 := div(2, 2)

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 = 1

Mobius roundtrip m=2 inverse F(n/2):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum := sum(15)

Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum = 15

Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) := mul(-1, 15)

Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) = -15

Mobius roundtrip m=2 inverse complete inverse-term sum := sum(34, -15)

Mobius roundtrip m=2 inverse complete inverse-term sum = 19

Mobius roundtrip m=2 inverse direct ORIGINAL a times n := mul(4, 2)

Mobius roundtrip m=2 inverse direct ORIGINAL a times n = 8

Mobius roundtrip m=2 inverse direct plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=2 inverse direct plus ORIGINAL b = 19

Mobius roundtrip m=2 inverse:actual n=2;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 2, 34, 34], [2, -1, 1, 15, -15]]

complete reconstructed=19;direct f(n)=19;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=3 forward:actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=3 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=3 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=3 forward f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=3 forward f(3) plus ORIGINAL b = 23

Mobius roundtrip m=3 forward full forward divisor sum := sum(15, 23)

Mobius roundtrip m=3 forward full forward divisor sum = 38

Mobius roundtrip m=3 inverse:actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 := div(3, 1)

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 = 3

Mobius roundtrip m=3 inverse F(n/1):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum := sum(15, 23)

Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum = 38

Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) := mul(1, 38)

Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) = 38

Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 := div(3, 3)

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 = 1

Mobius roundtrip m=3 inverse F(n/3):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum := sum(15)

Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum = 15

Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) := mul(-1, 15)

Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) = -15

Mobius roundtrip m=3 inverse complete inverse-term sum := sum(38, -15)

Mobius roundtrip m=3 inverse complete inverse-term sum = 23

Mobius roundtrip m=3 inverse direct ORIGINAL a times n := mul(4, 3)

Mobius roundtrip m=3 inverse direct ORIGINAL a times n = 12

Mobius roundtrip m=3 inverse direct plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=3 inverse direct plus ORIGINAL b = 23

Mobius roundtrip m=3 inverse:actual n=3;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 3, 38, 38], [3, -1, 1, 15, -15]]

complete reconstructed=23;direct f(n)=23;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=4 forward:actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=4 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=4 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=4 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=4 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=4 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=4 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=4 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=4 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=4 forward f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=4 forward f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=4 forward f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=4 forward f(4) plus ORIGINAL b = 27

Mobius roundtrip m=4 forward full forward divisor sum := sum(15, 19, 27)

Mobius roundtrip m=4 forward full forward divisor sum = 61

Mobius roundtrip m=4 inverse:actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor1 := div(4, 1)

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor1 = 4

Mobius roundtrip m=4 inverse F(n/1):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=4 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=4 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=4 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=4 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=4 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=4 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=4 inverse F(n/1) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=4 inverse F(n/1) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=4 inverse F(n/1) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=4 inverse F(n/1) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum := sum(15, 19, 27)

Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum = 61

Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1) := mul(1, 61)

Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1) = 61

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor2 := div(4, 2)

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor2 = 2

Mobius roundtrip m=4 inverse F(n/2):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=4 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=4 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=4 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=4 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=4 inverse F(n/2) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=4 inverse F(n/2) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum = 34

Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2) := mul(-1, 34)

Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2) = -34

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=4 inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor4 := div(4, 4)

Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor4 = 1

Mobius roundtrip m=4 inverse F(n/4):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=4 inverse F(n/4) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=4 inverse F(n/4) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=4 inverse F(n/4) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=4 inverse F(n/4) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=4 inverse F(n/4) full forward divisor sum := sum(15)

Mobius roundtrip m=4 inverse F(n/4) full forward divisor sum = 15

Mobius roundtrip m=4 inverse mu(4) times SAME F(n/4) := mul(0, 15)

Mobius roundtrip m=4 inverse mu(4) times SAME F(n/4) = 0

Mobius roundtrip m=4 inverse complete inverse-term sum := sum(61, -34, 0)

Mobius roundtrip m=4 inverse complete inverse-term sum = 27

Mobius roundtrip m=4 inverse direct ORIGINAL a times n := mul(4, 4)

Mobius roundtrip m=4 inverse direct ORIGINAL a times n = 16

Mobius roundtrip m=4 inverse direct plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=4 inverse direct plus ORIGINAL b = 27

Mobius roundtrip m=4 inverse:actual n=4;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 4, 61, 61], [2, -1, 2, 34, -34], [4, 0, 1, 15, 0]]

complete reconstructed=27;direct f(n)=27;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=6 forward:actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=6 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=6 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=6 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=6 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=6 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=6 forward f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=6 forward f(3) plus ORIGINAL b = 23

Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=6 forward f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=6 forward f(6) plus ORIGINAL b = 35

Mobius roundtrip m=6 forward full forward divisor sum := sum(15, 19, 23, 35)

Mobius roundtrip m=6 forward full forward divisor sum = 92

Mobius roundtrip m=6 inverse:actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1 := div(6, 1)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1 = 6

Mobius roundtrip m=6 inverse F(n/1):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum := sum(15, 19, 23, 35)

Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum = 92

Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1) := mul(1, 92)

Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1) = 92

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2 := div(6, 2)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2 = 3

Mobius roundtrip m=6 inverse F(n/2):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum := sum(15, 23)

Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum = 38

Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2) := mul(-1, 38)

Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2) = -38

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3 := div(6, 3)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3 = 2

Mobius roundtrip m=6 inverse F(n/3):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum = 34

Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3) := mul(-1, 34)

Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3) = -34

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=6>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=6 inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

mu=1;empty product case=否;repeated prime=否

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor6 := div(6, 6)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor6 = 1

Mobius roundtrip m=6 inverse F(n/6):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum := sum(15)

Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum = 15

Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6) := mul(1, 15)

Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6) = 15

Mobius roundtrip m=6 inverse complete inverse-term sum := sum(92, -38, -34, 15)

Mobius roundtrip m=6 inverse complete inverse-term sum = 35

Mobius roundtrip m=6 inverse direct ORIGINAL a times n := mul(4, 6)

Mobius roundtrip m=6 inverse direct ORIGINAL a times n = 24

Mobius roundtrip m=6 inverse direct plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=6 inverse direct plus ORIGINAL b = 35

Mobius roundtrip m=6 inverse:actual n=6;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 6, 92, 92], [2, -1, 3, 38, -38], [3, -1, 2, 34, -34], [6, 1, 1, 15, 15]]

complete reconstructed=35;direct f(n)=35;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=8 forward:actual positive input=8;check complete integer range1..8

ALL positive divisors INCREASING=[1, 2, 4, 8];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=8 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=8 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=8 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=8 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=8 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=8 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=8 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=8 forward f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=8 forward f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=8 forward f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=8 forward f(4) plus ORIGINAL b = 27

Mobius roundtrip m=8 forward f(8) ORIGINAL a times divisor := mul(4, 8)

Mobius roundtrip m=8 forward f(8) ORIGINAL a times divisor = 32

Mobius roundtrip m=8 forward f(8) plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=8 forward f(8) plus ORIGINAL b = 43

Mobius roundtrip m=8 forward full forward divisor sum := sum(15, 19, 27, 43)

Mobius roundtrip m=8 forward full forward divisor sum = 104

Mobius roundtrip m=8 inverse:actual positive input=8;check complete integer range1..8

ALL positive divisors INCREASING=[1, 2, 4, 8];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor1 := div(8, 1)

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor1 = 8

Mobius roundtrip m=8 inverse F(n/1):actual positive input=8;check complete integer range1..8

ALL positive divisors INCREASING=[1, 2, 4, 8];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=8 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=8 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=8 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=8 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=8 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=8 inverse F(n/1) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=8 inverse F(n/1) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=8 inverse F(n/1) f(8) ORIGINAL a times divisor := mul(4, 8)

Mobius roundtrip m=8 inverse F(n/1) f(8) ORIGINAL a times divisor = 32

Mobius roundtrip m=8 inverse F(n/1) f(8) plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=8 inverse F(n/1) f(8) plus ORIGINAL b = 43

Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum := sum(15, 19, 27, 43)

Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum = 104

Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1) := mul(1, 104)

Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1) = 104

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor2 := div(8, 2)

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor2 = 4

Mobius roundtrip m=8 inverse F(n/2):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=8 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=8 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=8 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=8 inverse F(n/2) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=8 inverse F(n/2) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=8 inverse F(n/2) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=8 inverse F(n/2) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=8 inverse F(n/2) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=8 inverse F(n/2) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum := sum(15, 19, 27)

Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum = 61

Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2) := mul(-1, 61)

Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2) = -61

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor4 := div(8, 4)

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor4 = 2

Mobius roundtrip m=8 inverse F(n/4):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 inverse F(n/4) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=8 inverse F(n/4) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=8 inverse F(n/4) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=8 inverse F(n/4) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=8 inverse F(n/4) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=8 inverse F(n/4) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum = 34

Mobius roundtrip m=8 inverse mu(4) times SAME F(n/4) := mul(0, 34)

Mobius roundtrip m=8 inverse mu(4) times SAME F(n/4) = 0

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor8:actual positive input for mu=8

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=8>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(8, 2)

upf extract divisor2 occurrence0 = 4

upf extract divisor2 occurrence1 := div(4, 2)

upf extract divisor2 occurrence1 = 2

upf extract divisor2 occurrence2 := div(2, 2)

upf extract divisor2 occurrence2 = 1

next divisor=2;previous remaining=8;all smaller divisors exhausted

extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8

complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=8 inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor8 := div(8, 8)

Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor8 = 1

Mobius roundtrip m=8 inverse F(n/8):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=8 inverse F(n/8) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=8 inverse F(n/8) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=8 inverse F(n/8) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=8 inverse F(n/8) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=8 inverse F(n/8) full forward divisor sum := sum(15)

Mobius roundtrip m=8 inverse F(n/8) full forward divisor sum = 15

Mobius roundtrip m=8 inverse mu(8) times SAME F(n/8) := mul(0, 15)

Mobius roundtrip m=8 inverse mu(8) times SAME F(n/8) = 0

Mobius roundtrip m=8 inverse complete inverse-term sum := sum(104, -61, 0, 0)

Mobius roundtrip m=8 inverse complete inverse-term sum = 43

Mobius roundtrip m=8 inverse direct ORIGINAL a times n := mul(4, 8)

Mobius roundtrip m=8 inverse direct ORIGINAL a times n = 32

Mobius roundtrip m=8 inverse direct plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=8 inverse direct plus ORIGINAL b = 43

Mobius roundtrip m=8 inverse:actual n=8;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 8, 104, 104], [2, -1, 4, 61, -61], [4, 0, 2, 34, 0], [8, 0, 1, 15, 0]]

complete reconstructed=43;direct f(n)=43;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=12 forward:actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=12 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=12 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=12 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=12 forward f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=12 forward f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=12 forward f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=12 forward f(3) plus ORIGINAL b = 23

Mobius roundtrip m=12 forward f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=12 forward f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=12 forward f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=12 forward f(4) plus ORIGINAL b = 27

Mobius roundtrip m=12 forward f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=12 forward f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=12 forward f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=12 forward f(6) plus ORIGINAL b = 35

Mobius roundtrip m=12 forward f(12) ORIGINAL a times divisor := mul(4, 12)

Mobius roundtrip m=12 forward f(12) ORIGINAL a times divisor = 48

Mobius roundtrip m=12 forward f(12) plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=12 forward f(12) plus ORIGINAL b = 59

Mobius roundtrip m=12 forward full forward divisor sum := sum(15, 19, 23, 27, 35, 59)

Mobius roundtrip m=12 forward full forward divisor sum = 178

Mobius roundtrip m=12 inverse:actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor1 := div(12, 1)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor1 = 12

Mobius roundtrip m=12 inverse F(n/1):actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=12 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=12 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=12 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=12 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=12 inverse F(n/1) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=12 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=12 inverse F(n/1) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=12 inverse F(n/1) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=12 inverse F(n/1) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=12 inverse F(n/1) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=12 inverse F(n/1) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=12 inverse F(n/1) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=12 inverse F(n/1) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=12 inverse F(n/1) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=12 inverse F(n/1) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=12 inverse F(n/1) f(12) ORIGINAL a times divisor := mul(4, 12)

Mobius roundtrip m=12 inverse F(n/1) f(12) ORIGINAL a times divisor = 48

Mobius roundtrip m=12 inverse F(n/1) f(12) plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=12 inverse F(n/1) f(12) plus ORIGINAL b = 59

Mobius roundtrip m=12 inverse F(n/1) full forward divisor sum := sum(15, 19, 23, 27, 35, 59)

Mobius roundtrip m=12 inverse F(n/1) full forward divisor sum = 178

Mobius roundtrip m=12 inverse mu(1) times SAME F(n/1) := mul(1, 178)

Mobius roundtrip m=12 inverse mu(1) times SAME F(n/1) = 178

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor2 := div(12, 2)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor2 = 6

Mobius roundtrip m=12 inverse F(n/2):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/2) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=12 inverse F(n/2) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=12 inverse F(n/2) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=12 inverse F(n/2) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=12 inverse F(n/2) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=12 inverse F(n/2) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=12 inverse F(n/2) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=12 inverse F(n/2) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=12 inverse F(n/2) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=12 inverse F(n/2) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=12 inverse F(n/2) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=12 inverse F(n/2) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=12 inverse F(n/2) full forward divisor sum := sum(15, 19, 23, 35)

Mobius roundtrip m=12 inverse F(n/2) full forward divisor sum = 92

Mobius roundtrip m=12 inverse mu(2) times SAME F(n/2) := mul(-1, 92)

Mobius roundtrip m=12 inverse mu(2) times SAME F(n/2) = -92

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor3 := div(12, 3)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor3 = 4

Mobius roundtrip m=12 inverse F(n/3):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/3) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/3) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/3) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=12 inverse F(n/3) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=12 inverse F(n/3) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=12 inverse F(n/3) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=12 inverse F(n/3) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=12 inverse F(n/3) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=12 inverse F(n/3) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=12 inverse F(n/3) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=12 inverse F(n/3) full forward divisor sum := sum(15, 19, 27)

Mobius roundtrip m=12 inverse F(n/3) full forward divisor sum = 61

Mobius roundtrip m=12 inverse mu(3) times SAME F(n/3) := mul(-1, 61)

Mobius roundtrip m=12 inverse mu(3) times SAME F(n/3) = -61

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor4 := div(12, 4)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor4 = 3

Mobius roundtrip m=12 inverse F(n/4):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/4) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/4) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/4) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/4) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/4) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=12 inverse F(n/4) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=12 inverse F(n/4) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=12 inverse F(n/4) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=12 inverse F(n/4) full forward divisor sum := sum(15, 23)

Mobius roundtrip m=12 inverse F(n/4) full forward divisor sum = 38

Mobius roundtrip m=12 inverse mu(4) times SAME F(n/4) := mul(0, 38)

Mobius roundtrip m=12 inverse mu(4) times SAME F(n/4) = 0

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=6>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

mu=1;empty product case=否;repeated prime=否

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor6 := div(12, 6)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor6 = 2

Mobius roundtrip m=12 inverse F(n/6):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/6) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/6) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/6) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/6) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/6) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=12 inverse F(n/6) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=12 inverse F(n/6) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=12 inverse F(n/6) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=12 inverse F(n/6) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=12 inverse F(n/6) full forward divisor sum = 34

Mobius roundtrip m=12 inverse mu(6) times SAME F(n/6) := mul(1, 34)

Mobius roundtrip m=12 inverse mu(6) times SAME F(n/6) = 34

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=12>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(12, 2)

upf extract divisor2 occurrence0 = 6

upf extract divisor2 occurrence1 := div(6, 2)

upf extract divisor2 occurrence1 = 3

next divisor=2;previous remaining=12;all smaller divisors exhausted

extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=12 inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor12 := div(12, 12)

Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor12 = 1

Mobius roundtrip m=12 inverse F(n/12):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=12 inverse F(n/12) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=12 inverse F(n/12) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=12 inverse F(n/12) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=12 inverse F(n/12) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=12 inverse F(n/12) full forward divisor sum := sum(15)

Mobius roundtrip m=12 inverse F(n/12) full forward divisor sum = 15

Mobius roundtrip m=12 inverse mu(12) times SAME F(n/12) := mul(0, 15)

Mobius roundtrip m=12 inverse mu(12) times SAME F(n/12) = 0

Mobius roundtrip m=12 inverse complete inverse-term sum := sum(178, -92, -61, 0, 34, 0)

Mobius roundtrip m=12 inverse complete inverse-term sum = 59

Mobius roundtrip m=12 inverse direct ORIGINAL a times n := mul(4, 12)

Mobius roundtrip m=12 inverse direct ORIGINAL a times n = 48

Mobius roundtrip m=12 inverse direct plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=12 inverse direct plus ORIGINAL b = 59

Mobius roundtrip m=12 inverse:actual n=12;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 12, 178, 178], [2, -1, 6, 92, -92], [3, -1, 4, 61, -61], [4, 0, 3, 38, 0], [6, 1, 2, 34, 34], [12, 0, 1, 15, 0]]

complete reconstructed=59;direct f(n)=59;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=24 forward:actual positive input=24;check complete integer range1..24

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 forward f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 forward f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 forward f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 forward f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 forward f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 forward f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 forward f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 forward f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 forward f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=24 forward f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=24 forward f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=24 forward f(3) plus ORIGINAL b = 23

Mobius roundtrip m=24 forward f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=24 forward f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=24 forward f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=24 forward f(4) plus ORIGINAL b = 27

Mobius roundtrip m=24 forward f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=24 forward f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=24 forward f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=24 forward f(6) plus ORIGINAL b = 35

Mobius roundtrip m=24 forward f(8) ORIGINAL a times divisor := mul(4, 8)

Mobius roundtrip m=24 forward f(8) ORIGINAL a times divisor = 32

Mobius roundtrip m=24 forward f(8) plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=24 forward f(8) plus ORIGINAL b = 43

Mobius roundtrip m=24 forward f(12) ORIGINAL a times divisor := mul(4, 12)

Mobius roundtrip m=24 forward f(12) ORIGINAL a times divisor = 48

Mobius roundtrip m=24 forward f(12) plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=24 forward f(12) plus ORIGINAL b = 59

Mobius roundtrip m=24 forward f(24) ORIGINAL a times divisor := mul(4, 24)

Mobius roundtrip m=24 forward f(24) ORIGINAL a times divisor = 96

Mobius roundtrip m=24 forward f(24) plus ORIGINAL b := sum(96, 11)

Mobius roundtrip m=24 forward f(24) plus ORIGINAL b = 107

Mobius roundtrip m=24 forward full forward divisor sum := sum(15, 19, 23, 27, 35, 43, 59, 107)

Mobius roundtrip m=24 forward full forward divisor sum = 328

Mobius roundtrip m=24 inverse:actual positive input=24;check complete integer range1..24

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=是;repeated prime=否

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor1 := div(24, 1)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor1 = 24

Mobius roundtrip m=24 inverse F(n/1):actual positive input=24;check complete integer range1..24

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/1) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/1) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/1) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/1) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/1) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=24 inverse F(n/1) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=24 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=24 inverse F(n/1) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=24 inverse F(n/1) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=24 inverse F(n/1) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=24 inverse F(n/1) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=24 inverse F(n/1) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=24 inverse F(n/1) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=24 inverse F(n/1) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=24 inverse F(n/1) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=24 inverse F(n/1) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=24 inverse F(n/1) f(8) ORIGINAL a times divisor := mul(4, 8)

Mobius roundtrip m=24 inverse F(n/1) f(8) ORIGINAL a times divisor = 32

Mobius roundtrip m=24 inverse F(n/1) f(8) plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=24 inverse F(n/1) f(8) plus ORIGINAL b = 43

Mobius roundtrip m=24 inverse F(n/1) f(12) ORIGINAL a times divisor := mul(4, 12)

Mobius roundtrip m=24 inverse F(n/1) f(12) ORIGINAL a times divisor = 48

Mobius roundtrip m=24 inverse F(n/1) f(12) plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=24 inverse F(n/1) f(12) plus ORIGINAL b = 59

Mobius roundtrip m=24 inverse F(n/1) f(24) ORIGINAL a times divisor := mul(4, 24)

Mobius roundtrip m=24 inverse F(n/1) f(24) ORIGINAL a times divisor = 96

Mobius roundtrip m=24 inverse F(n/1) f(24) plus ORIGINAL b := sum(96, 11)

Mobius roundtrip m=24 inverse F(n/1) f(24) plus ORIGINAL b = 107

Mobius roundtrip m=24 inverse F(n/1) full forward divisor sum := sum(15, 19, 23, 27, 35, 43, 59, 107)

Mobius roundtrip m=24 inverse F(n/1) full forward divisor sum = 328

Mobius roundtrip m=24 inverse mu(1) times SAME F(n/1) := mul(1, 328)

Mobius roundtrip m=24 inverse mu(1) times SAME F(n/1) = 328

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=2>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor2 := div(24, 2)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor2 = 12

Mobius roundtrip m=24 inverse F(n/2):actual positive input=12;check complete integer range1..12

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/2) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/2) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/2) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/2) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/2) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/2) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/2) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/2) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=24 inverse F(n/2) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=24 inverse F(n/2) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=24 inverse F(n/2) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=24 inverse F(n/2) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=24 inverse F(n/2) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=24 inverse F(n/2) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=24 inverse F(n/2) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=24 inverse F(n/2) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=24 inverse F(n/2) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=24 inverse F(n/2) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=24 inverse F(n/2) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=24 inverse F(n/2) f(12) ORIGINAL a times divisor := mul(4, 12)

Mobius roundtrip m=24 inverse F(n/2) f(12) ORIGINAL a times divisor = 48

Mobius roundtrip m=24 inverse F(n/2) f(12) plus ORIGINAL b := sum(48, 11)

Mobius roundtrip m=24 inverse F(n/2) f(12) plus ORIGINAL b = 59

Mobius roundtrip m=24 inverse F(n/2) full forward divisor sum := sum(15, 19, 23, 27, 35, 59)

Mobius roundtrip m=24 inverse F(n/2) full forward divisor sum = 178

Mobius roundtrip m=24 inverse mu(2) times SAME F(n/2) := mul(-1, 178)

Mobius roundtrip m=24 inverse mu(2) times SAME F(n/2) = -178

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=3>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

mu=-1;empty product case=否;repeated prime=否

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor3 := div(24, 3)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor3 = 8

Mobius roundtrip m=24 inverse F(n/3):actual positive input=8;check complete integer range1..8

ALL positive divisors INCREASING=[1, 2, 4, 8];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/3) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/3) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/3) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/3) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/3) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/3) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/3) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=24 inverse F(n/3) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=24 inverse F(n/3) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=24 inverse F(n/3) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=24 inverse F(n/3) f(8) ORIGINAL a times divisor := mul(4, 8)

Mobius roundtrip m=24 inverse F(n/3) f(8) ORIGINAL a times divisor = 32

Mobius roundtrip m=24 inverse F(n/3) f(8) plus ORIGINAL b := sum(32, 11)

Mobius roundtrip m=24 inverse F(n/3) f(8) plus ORIGINAL b = 43

Mobius roundtrip m=24 inverse F(n/3) full forward divisor sum := sum(15, 19, 27, 43)

Mobius roundtrip m=24 inverse F(n/3) full forward divisor sum = 104

Mobius roundtrip m=24 inverse mu(3) times SAME F(n/3) := mul(-1, 104)

Mobius roundtrip m=24 inverse mu(3) times SAME F(n/3) = -104

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=4>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor4 := div(24, 4)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor4 = 6

Mobius roundtrip m=24 inverse F(n/4):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/4) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/4) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/4) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/4) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/4) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/4) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/4) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/4) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/4) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=24 inverse F(n/4) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=24 inverse F(n/4) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=24 inverse F(n/4) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=24 inverse F(n/4) f(6) ORIGINAL a times divisor := mul(4, 6)

Mobius roundtrip m=24 inverse F(n/4) f(6) ORIGINAL a times divisor = 24

Mobius roundtrip m=24 inverse F(n/4) f(6) plus ORIGINAL b := sum(24, 11)

Mobius roundtrip m=24 inverse F(n/4) f(6) plus ORIGINAL b = 35

Mobius roundtrip m=24 inverse F(n/4) full forward divisor sum := sum(15, 19, 23, 35)

Mobius roundtrip m=24 inverse F(n/4) full forward divisor sum = 92

Mobius roundtrip m=24 inverse mu(4) times SAME F(n/4) := mul(0, 92)

Mobius roundtrip m=24 inverse mu(4) times SAME F(n/4) = 0

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=6>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

mu=1;empty product case=否;repeated prime=否

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor6 := div(24, 6)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor6 = 4

Mobius roundtrip m=24 inverse F(n/6):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/6) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/6) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/6) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/6) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/6) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/6) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/6) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/6) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/6) f(4) ORIGINAL a times divisor := mul(4, 4)

Mobius roundtrip m=24 inverse F(n/6) f(4) ORIGINAL a times divisor = 16

Mobius roundtrip m=24 inverse F(n/6) f(4) plus ORIGINAL b := sum(16, 11)

Mobius roundtrip m=24 inverse F(n/6) f(4) plus ORIGINAL b = 27

Mobius roundtrip m=24 inverse F(n/6) full forward divisor sum := sum(15, 19, 27)

Mobius roundtrip m=24 inverse F(n/6) full forward divisor sum = 61

Mobius roundtrip m=24 inverse mu(6) times SAME F(n/6) := mul(1, 61)

Mobius roundtrip m=24 inverse mu(6) times SAME F(n/6) = 61

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor8:actual positive input for mu=8

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=8>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(8, 2)

upf extract divisor2 occurrence0 = 4

upf extract divisor2 occurrence1 := div(4, 2)

upf extract divisor2 occurrence1 = 2

upf extract divisor2 occurrence2 := div(2, 2)

upf extract divisor2 occurrence2 = 1

next divisor=2;previous remaining=8;all smaller divisors exhausted

extracted multiplicity=3;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8

complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor8 := div(24, 8)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor8 = 3

Mobius roundtrip m=24 inverse F(n/8):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/8) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/8) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/8) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/8) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/8) f(3) ORIGINAL a times divisor := mul(4, 3)

Mobius roundtrip m=24 inverse F(n/8) f(3) ORIGINAL a times divisor = 12

Mobius roundtrip m=24 inverse F(n/8) f(3) plus ORIGINAL b := sum(12, 11)

Mobius roundtrip m=24 inverse F(n/8) f(3) plus ORIGINAL b = 23

Mobius roundtrip m=24 inverse F(n/8) full forward divisor sum := sum(15, 23)

Mobius roundtrip m=24 inverse F(n/8) full forward divisor sum = 38

Mobius roundtrip m=24 inverse mu(8) times SAME F(n/8) := mul(0, 38)

Mobius roundtrip m=24 inverse mu(8) times SAME F(n/8) = 0

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor12:actual positive input for mu=12

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=12>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(12, 2)

upf extract divisor2 occurrence0 = 6

upf extract divisor2 occurrence1 := div(6, 2)

upf extract divisor2 occurrence1 = 3

next divisor=2;previous remaining=12;all smaller divisors exhausted

extracted multiplicity=2;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor12:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor12 := div(24, 12)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor12 = 2

Mobius roundtrip m=24 inverse F(n/12):actual positive input=2;check complete integer range1..2

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/12) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/12) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/12) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/12) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/12) f(2) ORIGINAL a times divisor := mul(4, 2)

Mobius roundtrip m=24 inverse F(n/12) f(2) ORIGINAL a times divisor = 8

Mobius roundtrip m=24 inverse F(n/12) f(2) plus ORIGINAL b := sum(8, 11)

Mobius roundtrip m=24 inverse F(n/12) f(2) plus ORIGINAL b = 19

Mobius roundtrip m=24 inverse F(n/12) full forward divisor sum := sum(15, 19)

Mobius roundtrip m=24 inverse F(n/12) full forward divisor sum = 34

Mobius roundtrip m=24 inverse mu(12) times SAME F(n/12) := mul(0, 34)

Mobius roundtrip m=24 inverse mu(12) times SAME F(n/12) = 0

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor24:actual positive input for mu=24

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=24>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(24, 2)

upf extract divisor2 occurrence0 = 12

upf extract divisor2 occurrence1 := div(12, 2)

upf extract divisor2 occurrence1 = 6

upf extract divisor2 occurrence2 := div(6, 2)

upf extract divisor2 occurrence2 = 3

next divisor=2;previous remaining=24;all smaller divisors exhausted

extracted multiplicity=3;next remaining=3;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 3)

upf canonical complete product factor3 = 24

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=24

complete increasing prime multiset=[2, 2, 2, 3];full product=24;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=24 inverse Mobius at ORIGINAL divisor24:input=24;COMPLETE prime multiset=[2, 2, 2, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor24 := div(24, 24)

Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor24 = 1

Mobius roundtrip m=24 inverse F(n/24):actual positive input=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=24 inverse F(n/24) f(1) ORIGINAL a times divisor := mul(4, 1)

Mobius roundtrip m=24 inverse F(n/24) f(1) ORIGINAL a times divisor = 4

Mobius roundtrip m=24 inverse F(n/24) f(1) plus ORIGINAL b := sum(4, 11)

Mobius roundtrip m=24 inverse F(n/24) f(1) plus ORIGINAL b = 15

Mobius roundtrip m=24 inverse F(n/24) full forward divisor sum := sum(15)

Mobius roundtrip m=24 inverse F(n/24) full forward divisor sum = 15

Mobius roundtrip m=24 inverse mu(24) times SAME F(n/24) := mul(0, 15)

Mobius roundtrip m=24 inverse mu(24) times SAME F(n/24) = 0

Mobius roundtrip m=24 inverse complete inverse-term sum := sum(328, -178, -104, 0, 61, 0, 0, 0)

Mobius roundtrip m=24 inverse complete inverse-term sum = 107

Mobius roundtrip m=24 inverse direct ORIGINAL a times n := mul(4, 24)

Mobius roundtrip m=24 inverse direct ORIGINAL a times n = 96

Mobius roundtrip m=24 inverse direct plus ORIGINAL b := sum(96, 11)

Mobius roundtrip m=24 inverse direct plus ORIGINAL b = 107

Mobius roundtrip m=24 inverse:actual n=24;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]]

complete reconstructed=107;direct f(n)=107;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

逐步反向核對

• S578 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S577 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse direct plus ORIGINAL b，不以小數近似替代。
• S576 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse direct ORIGINAL a times n，不以小數近似替代。
• S575 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse complete inverse-term sum，不以小數近似替代。
• S574 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(24) times SAME F(n/24)，不以小數近似替代。
• S573 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/24) full forward divisor sum，不以小數近似替代。
• S572 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/24) f(1) plus ORIGINAL b，不以小數近似替代。
• S571 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/24) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S570 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S569 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor24，不以小數近似替代。
• S568 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S567 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S566 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S565 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S564 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S563 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S562 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S561 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S560 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S559 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S558 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S557 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S556 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(12) times SAME F(n/12)，不以小數近似替代。
• S555 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/12) full forward divisor sum，不以小數近似替代。
• S554 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/12) f(2) plus ORIGINAL b，不以小數近似替代。
• S553 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/12) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S552 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/12) f(1) plus ORIGINAL b，不以小數近似替代。
• S551 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/12) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S550 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S549 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor12，不以小數近似替代。
• S548 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S547 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S546 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S545 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S544 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S543 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S542 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S541 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S540 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S539 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S538 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(8) times SAME F(n/8)，不以小數近似替代。
• S537 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/8) full forward divisor sum，不以小數近似替代。
• S536 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/8) f(3) plus ORIGINAL b，不以小數近似替代。
• S535 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/8) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S534 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/8) f(1) plus ORIGINAL b，不以小數近似替代。
• S533 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/8) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S532 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S531 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor8，不以小數近似替代。
• S530 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S529 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S528 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S527 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S526 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S525 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S524 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S523 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S522 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S521 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S520 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S519 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S518 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S517 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(4) plus ORIGINAL b，不以小數近似替代。
• S516 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S515 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(2) plus ORIGINAL b，不以小數近似替代。
• S514 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S513 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S512 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S511 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S510 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S509 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S508 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S507 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S506 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S505 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S504 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S503 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S502 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S501 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S500 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S499 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(6) plus ORIGINAL b，不以小數近似替代。
• S498 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S497 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(3) plus ORIGINAL b，不以小數近似替代。
• S496 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S495 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(2) plus ORIGINAL b，不以小數近似替代。
• S494 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S493 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S492 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S491 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S490 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S489 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S488 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S487 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S486 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S485 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S484 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S483 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S482 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S481 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S480 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S479 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S478 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(8) plus ORIGINAL b，不以小數近似替代。
• S477 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S476 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(4) plus ORIGINAL b，不以小數近似替代。
• S475 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S474 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S473 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S472 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S471 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S470 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S469 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S468 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S467 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S466 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S465 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S464 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S463 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S462 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S461 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(12) plus ORIGINAL b，不以小數近似替代。
• S460 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S459 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(6) plus ORIGINAL b，不以小數近似替代。
• S458 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S457 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(4) plus ORIGINAL b，不以小數近似替代。
• S456 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S455 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S454 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S453 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S452 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S451 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S450 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S449 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S448 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S447 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S446 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S445 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S444 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S443 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S442 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S441 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S440 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(24) plus ORIGINAL b，不以小數近似替代。
• S439 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(24) ORIGINAL a times divisor，不以小數近似替代。
• S438 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(12) plus ORIGINAL b，不以小數近似替代。
• S437 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S436 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(8) plus ORIGINAL b，不以小數近似替代。
• S435 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S434 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S433 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S432 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S431 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S430 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S429 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S428 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S427 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S426 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S425 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S424 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S423 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S422 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S421 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S420 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S419 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward full forward divisor sum，不以小數近似替代。
• S418 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(24) plus ORIGINAL b，不以小數近似替代。
• S417 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(24) ORIGINAL a times divisor，不以小數近似替代。
• S416 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(12) plus ORIGINAL b，不以小數近似替代。
• S415 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(12) ORIGINAL a times divisor，不以小數近似替代。
• S414 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(8) plus ORIGINAL b，不以小數近似替代。
• S413 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(8) ORIGINAL a times divisor，不以小數近似替代。
• S412 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(6) plus ORIGINAL b，不以小數近似替代。
• S411 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(6) ORIGINAL a times divisor，不以小數近似替代。
• S410 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(4) plus ORIGINAL b，不以小數近似替代。
• S409 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(4) ORIGINAL a times divisor，不以小數近似替代。
• S408 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(3) plus ORIGINAL b，不以小數近似替代。
• S407 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(3) ORIGINAL a times divisor，不以小數近似替代。
• S406 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S405 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S404 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S403 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=24 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S402 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S401 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S400 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse direct plus ORIGINAL b，不以小數近似替代。
• S399 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse direct ORIGINAL a times n，不以小數近似替代。
• S398 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse complete inverse-term sum，不以小數近似替代。
• S397 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(12) times SAME F(n/12)，不以小數近似替代。
• S396 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/12) full forward divisor sum，不以小數近似替代。
• S395 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/12) f(1) plus ORIGINAL b，不以小數近似替代。
• S394 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/12) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S393 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S392 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor12，不以小數近似替代。
• S391 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S390 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S389 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S388 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S387 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S386 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S385 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S384 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S383 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S382 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S381 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S380 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S379 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/6) f(2) plus ORIGINAL b，不以小數近似替代。
• S378 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/6) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S377 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S376 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S375 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S374 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S373 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S372 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S371 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S370 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S369 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S368 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S367 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S366 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S365 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S364 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S363 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/4) f(3) plus ORIGINAL b，不以小數近似替代。
• S362 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/4) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S361 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S360 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S359 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S358 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S357 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S356 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S355 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S354 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S353 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S352 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S351 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S350 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S349 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S348 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S347 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S346 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(4) plus ORIGINAL b，不以小數近似替代。
• S345 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S344 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S343 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S342 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S341 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S340 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S339 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S338 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S337 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S336 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S335 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S334 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S333 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S332 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S331 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(6) plus ORIGINAL b，不以小數近似替代。
• S330 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S329 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S328 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S327 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S326 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S325 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S324 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S323 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S322 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S321 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S320 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S319 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S318 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S317 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S316 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S315 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S314 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(12) plus ORIGINAL b，不以小數近似替代。
• S313 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(12) ORIGINAL a times divisor，不以小數近似替代。
• S312 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S311 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S310 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S309 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S308 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S307 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S306 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S305 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S304 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S303 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S302 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S301 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S300 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S299 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S298 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S297 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward full forward divisor sum，不以小數近似替代。
• S296 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(12) plus ORIGINAL b，不以小數近似替代。
• S295 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(12) ORIGINAL a times divisor，不以小數近似替代。
• S294 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(6) plus ORIGINAL b，不以小數近似替代。
• S293 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(6) ORIGINAL a times divisor，不以小數近似替代。
• S292 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(4) plus ORIGINAL b，不以小數近似替代。
• S291 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(4) ORIGINAL a times divisor，不以小數近似替代。
• S290 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(3) plus ORIGINAL b，不以小數近似替代。
• S289 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(3) ORIGINAL a times divisor，不以小數近似替代。
• S288 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S287 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S286 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S285 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=12 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S284 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S283 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S282 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse direct plus ORIGINAL b，不以小數近似替代。
• S281 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse direct ORIGINAL a times n，不以小數近似替代。
• S280 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse complete inverse-term sum，不以小數近似替代。
• S279 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse mu(8) times SAME F(n/8)，不以小數近似替代。
• S278 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/8) full forward divisor sum，不以小數近似替代。
• S277 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/8) f(1) plus ORIGINAL b，不以小數近似替代。
• S276 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/8) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S275 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S274 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor8，不以小數近似替代。
• S273 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S272 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S271 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S270 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S269 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S268 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S267 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S266 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S265 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S264 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S263 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S262 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S261 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S260 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b，不以小數近似替代。
• S259 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/4) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S258 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S257 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S256 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S255 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S254 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S253 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S252 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S251 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S250 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S249 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S248 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S247 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S246 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S245 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S244 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S243 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b，不以小數近似替代。
• S242 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S241 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S240 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S239 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S238 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S237 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S236 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S235 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S234 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S233 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S232 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S231 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S230 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S229 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S228 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(8) plus ORIGINAL b，不以小數近似替代。
• S227 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(8) ORIGINAL a times divisor，不以小數近似替代。
• S226 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S225 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S224 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S223 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S222 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S221 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S220 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S219 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S218 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S217 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S216 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S215 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward full forward divisor sum，不以小數近似替代。
• S214 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(8) plus ORIGINAL b，不以小數近似替代。
• S213 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(8) ORIGINAL a times divisor，不以小數近似替代。
• S212 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(4) plus ORIGINAL b，不以小數近似替代。
• S211 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(4) ORIGINAL a times divisor，不以小數近似替代。
• S210 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S209 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S208 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S207 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=8 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S206 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S205 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S204 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse direct plus ORIGINAL b，不以小數近似替代。
• S203 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse direct ORIGINAL a times n，不以小數近似替代。
• S202 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse complete inverse-term sum，不以小數近似替代。
• S201 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S200 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S199 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S198 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S197 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S196 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S195 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S194 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S193 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S192 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S191 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S190 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S189 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S188 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S187 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S186 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S185 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S184 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S183 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S182 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S181 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S180 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S179 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S178 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S177 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S176 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S175 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S174 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S173 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S171 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S168 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S167 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S166 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S165 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S164 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S163 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S162 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S161 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S157 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S156 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S151 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S150 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S149 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S148 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S147 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S146 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward full forward divisor sum，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(6) plus ORIGINAL b，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(3) plus ORIGINAL b，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=6 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S137 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S136 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S135 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse direct plus ORIGINAL b，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse direct ORIGINAL a times n，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse complete inverse-term sum，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse mu(4) times SAME F(n/4)，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/4) full forward divisor sum，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/4) f(1) plus ORIGINAL b，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/4) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S128 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S127 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor4，不以小數近似替代。
• S126 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S125 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S124 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S122 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S121 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S119 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S118 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S117 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/2) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S111 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S110 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S109 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S108 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S107 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S106 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S105 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S104 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(4) plus ORIGINAL b，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(4) ORIGINAL a times divisor，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S096 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S095 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S094 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S093 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S092 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S091 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward full forward divisor sum，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(4) plus ORIGINAL b，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(4) ORIGINAL a times divisor，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S087 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=4 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S084 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S083 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S082 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse direct plus ORIGINAL b，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse direct ORIGINAL a times n，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse complete inverse-term sum，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S073 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S072 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S071 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S070 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S068 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S062 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S061 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S057 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 forward full forward divisor sum，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 forward f(3) plus ORIGINAL b，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S050 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse direct plus ORIGINAL b，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse direct ORIGINAL a times n，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse complete inverse-term sum，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S039 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 forward full forward divisor sum，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 forward f(2) plus ORIGINAL b，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse direct plus ORIGINAL b，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse direct ORIGINAL a times n，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse complete inverse-term sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 forward full forward divisor sum，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 forward f(1) plus ORIGINAL b，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], 是, finite_divisor_poset_roundtrip)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。