Question 22191: Every Divisor Row Recovers Its Original Arithmetic Value

Original question and givens

Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

• G001 — Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-mobius-inversion-finite-divisors

Step-by-step derivation

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=16;check complete integer range1..16

ALL positive divisors INCREASING=[1, 2, 4, 8, 16];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(1, 1)

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

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

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

Mobius roundtrip m=1 forward full forward divisor sum := sum(4)

Mobius roundtrip m=1 forward full forward divisor sum = 4

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=Yes;repeated prime=No

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(1, 1)

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

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

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

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

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

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

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

Mobius roundtrip m=1 inverse complete inverse-term sum := sum(4)

Mobius roundtrip m=1 inverse complete inverse-term sum = 4

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

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

Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(1, 3)

Mobius roundtrip m=1 inverse direct plus ORIGINAL b = 4

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, 4, 4]]

complete reconstructed=4;direct f(n)=4;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(1, 1)

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

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b := sum(1, 3)

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b = 4

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

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

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b = 5

Mobius roundtrip m=2 forward full forward divisor sum := sum(4, 5)

Mobius roundtrip m=2 forward full forward divisor sum = 9

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=Yes;repeated prime=No

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(1, 1)

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

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

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

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

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

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

Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b = 5

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum := sum(4, 5)

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum = 9

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) := mul(1, 9)

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) = 9

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=No;repeated prime=No

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(1, 1)

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

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

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

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

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

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

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

Mobius roundtrip m=2 inverse complete inverse-term sum := sum(9, -4)

Mobius roundtrip m=2 inverse complete inverse-term sum = 5

Mobius roundtrip m=2 inverse direct ORIGINAL a times n := mul(1, 2)

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

Mobius roundtrip m=2 inverse direct plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=2 inverse direct plus ORIGINAL b = 5

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, 9, 9], [2, -1, 1, 4, -4]]

complete reconstructed=5;direct f(n)=5;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(1, 1)

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

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

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

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

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

Mobius roundtrip m=4 forward f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=4 forward f(2) plus ORIGINAL b = 5

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

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

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

Mobius roundtrip m=4 forward f(4) plus ORIGINAL b = 7

Mobius roundtrip m=4 forward full forward divisor sum := sum(4, 5, 7)

Mobius roundtrip m=4 forward full forward divisor sum = 16

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=Yes;repeated prime=No

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(1, 1)

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

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

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

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

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

Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b = 5

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

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

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

Mobius roundtrip m=4 inverse F(n/1) f(4) plus ORIGINAL b = 7

Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum := sum(4, 5, 7)

Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum = 16

Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1) := mul(1, 16)

Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1) = 16

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=No;repeated prime=No

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(1, 1)

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

Mobius roundtrip m=4 inverse F(n/2) f(1) plus ORIGINAL b := sum(1, 3)

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

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

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

Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b = 5

Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum := sum(4, 5)

Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum = 9

Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2) := mul(-1, 9)

Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2) = -9

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=No;repeated prime=Yes

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(1, 1)

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

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

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

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

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

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

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

Mobius roundtrip m=4 inverse complete inverse-term sum := sum(16, -9, 0)

Mobius roundtrip m=4 inverse complete inverse-term sum = 7

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

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

Mobius roundtrip m=4 inverse direct plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=4 inverse direct plus ORIGINAL b = 7

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, 16, 16], [2, -1, 2, 9, -9], [4, 0, 1, 4, 0]]

complete reconstructed=7;direct f(n)=7;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(1, 1)

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

Mobius roundtrip m=8 forward f(1) plus ORIGINAL b := sum(1, 3)

Mobius roundtrip m=8 forward f(1) plus ORIGINAL b = 4

Mobius roundtrip m=8 forward f(2) ORIGINAL a times divisor := mul(1, 2)

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

Mobius roundtrip m=8 forward f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=8 forward f(2) plus ORIGINAL b = 5

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

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

Mobius roundtrip m=8 forward f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=8 forward f(4) plus ORIGINAL b = 7

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

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

Mobius roundtrip m=8 forward f(8) plus ORIGINAL b := sum(8, 3)

Mobius roundtrip m=8 forward f(8) plus ORIGINAL b = 11

Mobius roundtrip m=8 forward full forward divisor sum := sum(4, 5, 7, 11)

Mobius roundtrip m=8 forward full forward divisor sum = 27

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=Yes;repeated prime=No

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(1, 1)

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

Mobius roundtrip m=8 inverse F(n/1) f(1) plus ORIGINAL b := sum(1, 3)

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

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

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

Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b = 5

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

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

Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b = 7

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

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

Mobius roundtrip m=8 inverse F(n/1) f(8) plus ORIGINAL b := sum(8, 3)

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

Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum := sum(4, 5, 7, 11)

Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum = 27

Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1) := mul(1, 27)

Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1) = 27

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=No;repeated prime=No

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(1, 1)

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

Mobius roundtrip m=8 inverse F(n/2) f(1) plus ORIGINAL b := sum(1, 3)

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

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

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

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

Mobius roundtrip m=8 inverse F(n/2) f(2) plus ORIGINAL b = 5

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

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

Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b = 7

Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum := sum(4, 5, 7)

Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum = 16

Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2) := mul(-1, 16)

Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2) = -16

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=No;repeated prime=Yes

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(1, 1)

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

Mobius roundtrip m=8 inverse F(n/4) f(1) plus ORIGINAL b := sum(1, 3)

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

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

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

Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b = 5

Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum := sum(4, 5)

Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum = 9

Mobius roundtrip m=8 inverse mu(4) times SAME F(n/4) := mul(0, 9)

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=No;repeated prime=Yes

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(1, 1)

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

Mobius roundtrip m=8 inverse F(n/8) f(1) plus ORIGINAL b := sum(1, 3)

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

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

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

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

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

Mobius roundtrip m=8 inverse complete inverse-term sum := sum(27, -16, 0, 0)

Mobius roundtrip m=8 inverse complete inverse-term sum = 11

Mobius roundtrip m=8 inverse direct ORIGINAL a times n := mul(1, 8)

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

Mobius roundtrip m=8 inverse direct plus ORIGINAL b := sum(8, 3)

Mobius roundtrip m=8 inverse direct plus ORIGINAL b = 11

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, 27, 27], [2, -1, 4, 16, -16], [4, 0, 2, 9, 0], [8, 0, 1, 4, 0]]

complete reconstructed=11;direct f(n)=11;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=16 forward:actual positive input=16;check complete integer range1..16

ALL positive divisors INCREASING=[1, 2, 4, 8, 16];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=16 forward f(1) ORIGINAL a times divisor := mul(1, 1)

Mobius roundtrip m=16 forward f(1) ORIGINAL a times divisor = 1

Mobius roundtrip m=16 forward f(1) plus ORIGINAL b := sum(1, 3)

Mobius roundtrip m=16 forward f(1) plus ORIGINAL b = 4

Mobius roundtrip m=16 forward f(2) ORIGINAL a times divisor := mul(1, 2)

Mobius roundtrip m=16 forward f(2) ORIGINAL a times divisor = 2

Mobius roundtrip m=16 forward f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=16 forward f(2) plus ORIGINAL b = 5

Mobius roundtrip m=16 forward f(4) ORIGINAL a times divisor := mul(1, 4)

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

Mobius roundtrip m=16 forward f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=16 forward f(4) plus ORIGINAL b = 7

Mobius roundtrip m=16 forward f(8) ORIGINAL a times divisor := mul(1, 8)

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

Mobius roundtrip m=16 forward f(8) plus ORIGINAL b := sum(8, 3)

Mobius roundtrip m=16 forward f(8) plus ORIGINAL b = 11

Mobius roundtrip m=16 forward f(16) ORIGINAL a times divisor := mul(1, 16)

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

Mobius roundtrip m=16 forward f(16) plus ORIGINAL b := sum(16, 3)

Mobius roundtrip m=16 forward f(16) plus ORIGINAL b = 19

Mobius roundtrip m=16 forward full forward divisor sum := sum(4, 5, 7, 11, 19)

Mobius roundtrip m=16 forward full forward divisor sum = 46

Mobius roundtrip m=16 inverse:actual positive input=16;check complete integer range1..16

ALL positive divisors INCREASING=[1, 2, 4, 8, 16];not only primes/proper divisors or all integers<=n

Mobius roundtrip m=16 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=16 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

mu=1;empty product case=Yes;repeated prime=No

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor1 := div(16, 1)

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor1 = 16

Mobius roundtrip m=16 inverse F(n/1):actual positive input=16;check complete integer range1..16

ALL positive divisors INCREASING=[1, 2, 4, 8, 16];not only primes/proper divisors or all integers<=n

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

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

Mobius roundtrip m=16 inverse F(n/1) f(1) plus ORIGINAL b := sum(1, 3)

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

Mobius roundtrip m=16 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(1, 2)

Mobius roundtrip m=16 inverse F(n/1) f(2) ORIGINAL a times divisor = 2

Mobius roundtrip m=16 inverse F(n/1) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=16 inverse F(n/1) f(2) plus ORIGINAL b = 5

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

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

Mobius roundtrip m=16 inverse F(n/1) f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=16 inverse F(n/1) f(4) plus ORIGINAL b = 7

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

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

Mobius roundtrip m=16 inverse F(n/1) f(8) plus ORIGINAL b := sum(8, 3)

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

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

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

Mobius roundtrip m=16 inverse F(n/1) f(16) plus ORIGINAL b := sum(16, 3)

Mobius roundtrip m=16 inverse F(n/1) f(16) plus ORIGINAL b = 19

Mobius roundtrip m=16 inverse F(n/1) full forward divisor sum := sum(4, 5, 7, 11, 19)

Mobius roundtrip m=16 inverse F(n/1) full forward divisor sum = 46

Mobius roundtrip m=16 inverse mu(1) times SAME F(n/1) := mul(1, 46)

Mobius roundtrip m=16 inverse mu(1) times SAME F(n/1) = 46

Mobius roundtrip m=16 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=16 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

mu=-1;empty product case=No;repeated prime=No

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor2 := div(16, 2)

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor2 = 8

Mobius roundtrip m=16 inverse F(n/2):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=16 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(1, 1)

Mobius roundtrip m=16 inverse F(n/2) f(1) ORIGINAL a times divisor = 1

Mobius roundtrip m=16 inverse F(n/2) f(1) plus ORIGINAL b := sum(1, 3)

Mobius roundtrip m=16 inverse F(n/2) f(1) plus ORIGINAL b = 4

Mobius roundtrip m=16 inverse F(n/2) f(2) ORIGINAL a times divisor := mul(1, 2)

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

Mobius roundtrip m=16 inverse F(n/2) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=16 inverse F(n/2) f(2) plus ORIGINAL b = 5

Mobius roundtrip m=16 inverse F(n/2) f(4) ORIGINAL a times divisor := mul(1, 4)

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

Mobius roundtrip m=16 inverse F(n/2) f(4) plus ORIGINAL b := sum(4, 3)

Mobius roundtrip m=16 inverse F(n/2) f(4) plus ORIGINAL b = 7

Mobius roundtrip m=16 inverse F(n/2) f(8) ORIGINAL a times divisor := mul(1, 8)

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

Mobius roundtrip m=16 inverse F(n/2) f(8) plus ORIGINAL b := sum(8, 3)

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

Mobius roundtrip m=16 inverse F(n/2) full forward divisor sum := sum(4, 5, 7, 11)

Mobius roundtrip m=16 inverse F(n/2) full forward divisor sum = 27

Mobius roundtrip m=16 inverse mu(2) times SAME F(n/2) := mul(-1, 27)

Mobius roundtrip m=16 inverse mu(2) times SAME F(n/2) = -27

Mobius roundtrip m=16 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=16 inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

mu=0;empty product case=No;repeated prime=Yes

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor4 := div(16, 4)

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor4 = 4

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

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

Mobius roundtrip m=16 inverse F(n/4) f(1) plus ORIGINAL b := sum(1, 3)

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

Mobius roundtrip m=16 inverse F(n/4) f(2) ORIGINAL a times divisor := mul(1, 2)

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

Mobius roundtrip m=16 inverse F(n/4) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=16 inverse F(n/4) f(2) plus ORIGINAL b = 5

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

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

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

Mobius roundtrip m=16 inverse F(n/4) f(4) plus ORIGINAL b = 7

Mobius roundtrip m=16 inverse F(n/4) full forward divisor sum := sum(4, 5, 7)

Mobius roundtrip m=16 inverse F(n/4) full forward divisor sum = 16

Mobius roundtrip m=16 inverse mu(4) times SAME F(n/4) := mul(0, 16)

Mobius roundtrip m=16 inverse mu(4) times SAME F(n/4) = 0

Mobius roundtrip m=16 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=16 inverse Mobius at ORIGINAL divisor8:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

mu=0;empty product case=No;repeated prime=Yes

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor8 := div(16, 8)

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor8 = 2

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

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

Mobius roundtrip m=16 inverse F(n/8) f(1) plus ORIGINAL b := sum(1, 3)

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

Mobius roundtrip m=16 inverse F(n/8) f(2) ORIGINAL a times divisor := mul(1, 2)

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

Mobius roundtrip m=16 inverse F(n/8) f(2) plus ORIGINAL b := sum(2, 3)

Mobius roundtrip m=16 inverse F(n/8) f(2) plus ORIGINAL b = 5

Mobius roundtrip m=16 inverse F(n/8) full forward divisor sum := sum(4, 5)

Mobius roundtrip m=16 inverse F(n/8) full forward divisor sum = 9

Mobius roundtrip m=16 inverse mu(8) times SAME F(n/8) := mul(0, 9)

Mobius roundtrip m=16 inverse mu(8) times SAME F(n/8) = 0

Mobius roundtrip m=16 inverse Mobius at ORIGINAL divisor16:actual positive input for mu=16

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

ORIGINAL positive integer absolute magnitude=16>1

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

upf extract divisor2 occurrence0 := div(16, 2)

upf extract divisor2 occurrence0 = 8

upf extract divisor2 occurrence1 := div(8, 2)

upf extract divisor2 occurrence1 = 4

upf extract divisor2 occurrence2 := div(4, 2)

upf extract divisor2 occurrence2 = 2

upf extract divisor2 occurrence3 := div(2, 2)

upf extract divisor2 occurrence3 = 1

next divisor=2;previous remaining=16;all smaller divisors exhausted

extracted multiplicity=4;next remaining=1;remaining modulo2=1;extracted multiset=[2, 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, 2)

upf canonical complete product factor3 = 16

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=16

complete increasing prime multiset=[2, 2, 2, 2];full product=16;a composite terminal remainder would have an already tested smaller divisor

Mobius roundtrip m=16 inverse Mobius at ORIGINAL divisor16:input=16;COMPLETE prime multiset=[2, 2, 2, 2];distinct primes=[2]

mu=0;empty product case=No;repeated prime=Yes

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor16 := div(16, 16)

Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor16 = 1

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

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

Mobius roundtrip m=16 inverse F(n/16) f(1) plus ORIGINAL b := sum(1, 3)

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

Mobius roundtrip m=16 inverse F(n/16) full forward divisor sum := sum(4)

Mobius roundtrip m=16 inverse F(n/16) full forward divisor sum = 4

Mobius roundtrip m=16 inverse mu(16) times SAME F(n/16) := mul(0, 4)

Mobius roundtrip m=16 inverse mu(16) times SAME F(n/16) = 0

Mobius roundtrip m=16 inverse complete inverse-term sum := sum(46, -27, 0, 0, 0)

Mobius roundtrip m=16 inverse complete inverse-term sum = 19

Mobius roundtrip m=16 inverse direct ORIGINAL a times n := mul(1, 16)

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

Mobius roundtrip m=16 inverse direct plus ORIGINAL b := sum(16, 3)

Mobius roundtrip m=16 inverse direct plus ORIGINAL b = 19

Mobius roundtrip m=16 inverse:actual n=16;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]]

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)

Reverse checks

• S289 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S288 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S287 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S286 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S285 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse mu(16) times SAME F(n/16) with exact rational arithmetic, not rounded decimals.
• S284 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/16) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/16) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/16) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S280 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor16 with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S278 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S277 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S276 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S275 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S274 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S272 · premise_and_result_audit — Recompute upf extract divisor2 occurrence3 with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S266 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse mu(8) times SAME F(n/8) with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/8) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/8) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/8) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/8) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/8) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S260 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S259 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor8 with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S257 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S256 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S253 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S252 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S250 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S248 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S247 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S238 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S235 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S228 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S217 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S214 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S211 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S210 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(16) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(16) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S206 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S198 · premise_and_result_audit — Recompute Mobius roundtrip m=16 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S194 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(16) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(16) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S188 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute Mobius roundtrip m=16 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S181 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse mu(8) times SAME F(n/8) with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/8) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/8) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/8) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S173 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor8 with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S166 · premise_and_result_audit — Recompute upf extract divisor2 occurrence2 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/4) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/4) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Recompute Mobius roundtrip m=8 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Mobius roundtrip m=8 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Recompute Mobius roundtrip m=4 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Mobius roundtrip m=4 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], Yes, finite_divisor_poset_roundtrip)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.