第 22130 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-incomplete-divisor-boundary；資料={"n":42,"a":5,"b":1,"omitted_divisor":1}。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-incomplete-divisor-boundary；資料={"n":42,"a":5,"b":1,"omitted_divisor":1}。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-incomplete-divisor-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]

Mobius inversion ORIGINAL inverse:actual positive input=42;check complete integer range1..42

ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 14, 21, 42];not only primes/proper divisors or all integers<=n

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(42, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 42

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=42;check complete integer range1..42

ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 14, 21, 42];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(5, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(10, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(5, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 15

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(15, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 16

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(5, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 30

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(30, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 31

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor := mul(5, 7)

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 35

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(35, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 36

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor := mul(5, 14)

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 70

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b := sum(70, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 71

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor := mul(5, 21)

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 105

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b := sum(105, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 106

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor := mul(5, 42)

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 210

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b := sum(210, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 211

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(6, 11, 16, 31, 36, 71, 106, 211)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 488

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 488)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 488

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(42, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 21

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=21;check complete integer range1..21

ALL positive divisors INCREASING=[1, 3, 7, 21];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(5, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 15

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(15, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 16

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(5, 7)

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 35

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(35, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 36

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor := mul(5, 21)

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 105

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b := sum(105, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 106

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(6, 16, 36, 106)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 164

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 164)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -164

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(42, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 14

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=14;check complete integer range1..14

ALL positive divisors INCREASING=[1, 2, 7, 14];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(5, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(10, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor := mul(5, 7)

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 35

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b := sum(35, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 36

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(5, 14)

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 70

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b := sum(70, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 71

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(6, 11, 36, 71)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 124

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 124)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -124

Mobius inversion ORIGINAL 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 inversion ORIGINAL 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 inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(42, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 7

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=7;check complete integer range1..7

ALL positive divisors INCREASING=[1, 7];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(5, 7)

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 35

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(35, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 36

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(6, 36)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 42

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 42)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 42

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7

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

ORIGINAL positive integer absolute magnitude=7>1

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

next divisor=2;previous remaining=7;all smaller divisors exhausted

extracted multiplicity=0;next remaining=7;remaining modulo2=1;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 7)

upf canonical complete product factor0 = 7

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7

complete increasing prime multiset=[7];full product=7;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(42, 7)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 6

Mobius inversion ORIGINAL inverse F(n/7):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 inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(5, 2)

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(10, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(5, 3)

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 15

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(15, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 16

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor := mul(5, 6)

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 30

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b := sum(30, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 31

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(6, 11, 16, 31)

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 64

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 64)

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -64

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14

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

ORIGINAL positive integer absolute magnitude=14>1

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

upf extract divisor2 occurrence0 := div(14, 2)

upf extract divisor2 occurrence0 = 7

next divisor=2;previous remaining=14;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;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, 7)

upf canonical complete product factor1 = 14

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14

complete increasing prime multiset=[2, 7];full product=14;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(42, 14)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 3

Mobius inversion ORIGINAL inverse F(n/14):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 inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(5, 3)

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 15

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(15, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 16

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(6, 16)

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 22

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 22)

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 22

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:actual positive input for mu=21

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

ORIGINAL positive integer absolute magnitude=21>1

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

next divisor=2;previous remaining=21;all smaller divisors exhausted

extracted multiplicity=0;next remaining=21;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(21, 3)

upf extract divisor3 occurrence0 = 7

next divisor=3;previous remaining=21;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[3]

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

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 7)

upf canonical complete product factor1 = 21

exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=21

complete increasing prime multiset=[3, 7];full product=21;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:input=21;COMPLETE prime multiset=[3, 7];distinct primes=[3, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 := div(42, 21)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 2

Mobius inversion ORIGINAL inverse F(n/21):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 inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(5, 2)

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(10, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 11

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum := sum(6, 11)

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 17

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) := mul(1, 17)

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 17

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:actual positive input for mu=42

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

ORIGINAL positive integer absolute magnitude=42>1

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

upf extract divisor2 occurrence0 := div(42, 2)

upf extract divisor2 occurrence0 = 21

next divisor=2;previous remaining=42;all smaller divisors exhausted

extracted multiplicity=1;next remaining=21;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(21, 3)

upf extract divisor3 occurrence1 = 7

next divisor=3;previous remaining=21;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 3]

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

upf canonical complete product factor2 := mul(6, 7)

upf canonical complete product factor2 = 42

exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=42

complete increasing prime multiset=[2, 3, 7];full product=42;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:input=42;COMPLETE prime multiset=[2, 3, 7];distinct primes=[2, 3, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 := div(42, 42)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 1

Mobius inversion ORIGINAL inverse F(n/42):actual positive input=1;check complete integer range1..1

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

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(5, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 6

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(6)

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 6

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 6)

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -6

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(488, -164, -124, 42, -64, 22, 17, -6)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 211

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(5, 42)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 210

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(210, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 211

Mobius inversion ORIGINAL inverse:actual n=42;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 42, 488, 488], [2, -1, 21, 164, -164], [3, -1, 14, 124, -124], [6, 1, 7, 42, 42], [7, -1, 6, 64, -64], [14, 1, 3, 22, 22], [21, 1, 2, 17, 17], [42, -1, 1, 6, -6]]

complete reconstructed=211;direct f(n)=211;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-164, -124, 42, -64, 22, 17, -6)

Mobius inversion ORIGINAL omitted-divisor partial sum = -277

Mobius inversion complete minus partial := sub(211, -277)

Mobius inversion complete minus partial = 488

ORIGINAL full divisors=[1, 2, 3, 6, 7, 14, 21, 42];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 7, 14, 21, 42]

complete=211=f(n)=211;partial=-277;lost term=488!=0;incomplete_certificate fixed No

逐步反向核對

• 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 — 以精確有理數重算 Mobius inversion complete minus partial，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL omitted-divisor partial sum，不以小數近似替代。
• S146 · 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。
• S145 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct 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 inversion ORIGINAL inverse direct ORIGINAL a times n，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse complete inverse-term sum，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42)，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S138 · 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。
• S137 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S134 · 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。
• S133 · 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。
• S132 · 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。
• S131 · 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。
• S130 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S129 · 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。
• S128 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S127 · 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。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21)，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b，不以小數近似替代。
• S122 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor，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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21，不以小數近似替代。
• S117 · 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。
• S116 · 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。
• S115 · 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。
• S114 · 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。
• S113 · 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。
• S112 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14)，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S102 · 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。
• S101 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14，不以小數近似替代。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• 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 upf extract divisor2 occurrence0，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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7)，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) 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 inversion ORIGINAL inverse F(n/7) f(6) 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 inversion ORIGINAL inverse F(n/7) f(6) 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 inversion ORIGINAL inverse F(n/7) f(3) 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 inversion ORIGINAL inverse F(n/7) f(3) 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 inversion ORIGINAL inverse F(n/7) f(2) 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 inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S082 · 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。
• S081 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7，不以小數近似替代。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• 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 inversion ORIGINAL inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S068 · 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。
• S067 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• 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 upf extract divisor2 occurrence0，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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) 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 inversion ORIGINAL inverse F(n/3) f(14) 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 inversion ORIGINAL inverse F(n/3) f(14) 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 inversion ORIGINAL inverse F(n/3) f(7) 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 inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) 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 inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S029 · 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。
• 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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(42) 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 inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(14) 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 inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S006 · 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S004 · 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。
• 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

(42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

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。