Question 22136: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=156, exact integer arithmetic function f(k)=a*k+b with a=11, b=1. 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. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

• G001 — Use positive integer n=156, exact integer arithmetic function f(k)=a*k+b with a=11, b=1. 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. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, 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-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=156;check complete integer range1..156

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156];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=Yes;repeated prime=No

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(156, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 156

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=156;check complete integer range1..156

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(11, 4)

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(11, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor := mul(11, 12)

Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor := mul(11, 26)

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 286

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(286, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 287

Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor := mul(11, 39)

Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor = 429

Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b := sum(429, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b = 430

Mobius inversion ORIGINAL inverse F(n/1) f(52) ORIGINAL a times divisor := mul(11, 52)

Mobius inversion ORIGINAL inverse F(n/1) f(52) ORIGINAL a times divisor = 572

Mobius inversion ORIGINAL inverse F(n/1) f(52) plus ORIGINAL b := sum(572, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(52) plus ORIGINAL b = 573

Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor := mul(11, 78)

Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor = 858

Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b := sum(858, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b = 859

Mobius inversion ORIGINAL inverse F(n/1) f(156) ORIGINAL a times divisor := mul(11, 156)

Mobius inversion ORIGINAL inverse F(n/1) f(156) ORIGINAL a times divisor = 1716

Mobius inversion ORIGINAL inverse F(n/1) f(156) plus ORIGINAL b := sum(1716, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(156) plus ORIGINAL b = 1717

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(12, 23, 34, 45, 67, 133, 144, 287, 430, 573, 859, 1717)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 4324

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 4324)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 4324

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=No;repeated prime=No

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(156, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 78

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=78;check complete integer range1..78

ALL positive divisors INCREASING=[1, 2, 3, 6, 13, 26, 39, 78];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor := mul(11, 6)

Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/2) f(26) ORIGINAL a times divisor := mul(11, 26)

Mobius inversion ORIGINAL inverse F(n/2) f(26) ORIGINAL a times divisor = 286

Mobius inversion ORIGINAL inverse F(n/2) f(26) plus ORIGINAL b := sum(286, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(26) plus ORIGINAL b = 287

Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor := mul(11, 39)

Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor = 429

Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b := sum(429, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b = 430

Mobius inversion ORIGINAL inverse F(n/2) f(78) ORIGINAL a times divisor := mul(11, 78)

Mobius inversion ORIGINAL inverse F(n/2) f(78) ORIGINAL a times divisor = 858

Mobius inversion ORIGINAL inverse F(n/2) f(78) plus ORIGINAL b := sum(858, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(78) plus ORIGINAL b = 859

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(12, 23, 34, 67, 144, 287, 430, 859)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 1856

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 1856)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -1856

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=No;repeated prime=No

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(156, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 52

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=52;check complete integer range1..52

ALL positive divisors INCREASING=[1, 2, 4, 13, 26, 52];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor := mul(11, 4)

Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor := mul(11, 26)

Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor = 286

Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b := sum(286, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b = 287

Mobius inversion ORIGINAL inverse F(n/3) f(52) ORIGINAL a times divisor := mul(11, 52)

Mobius inversion ORIGINAL inverse F(n/3) f(52) ORIGINAL a times divisor = 572

Mobius inversion ORIGINAL inverse F(n/3) f(52) plus ORIGINAL b := sum(572, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(52) plus ORIGINAL b = 573

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(12, 23, 45, 144, 287, 573)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 1084

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 1084)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -1084

Mobius inversion ORIGINAL 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 inversion ORIGINAL 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 inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(156, 4)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 39

Mobius inversion ORIGINAL inverse F(n/4):actual positive input=39;check complete integer range1..39

ALL positive divisors INCREASING=[1, 3, 13, 39];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/4) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/4) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/4) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/4) f(39) ORIGINAL a times divisor := mul(11, 39)

Mobius inversion ORIGINAL inverse F(n/4) f(39) ORIGINAL a times divisor = 429

Mobius inversion ORIGINAL inverse F(n/4) f(39) plus ORIGINAL b := sum(429, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(39) plus ORIGINAL b = 430

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(12, 34, 144, 430)

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 620

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 620)

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0

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=No;repeated prime=No

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(156, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 26

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=26;check complete integer range1..26

ALL positive divisors INCREASING=[1, 2, 13, 26];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/6) f(26) ORIGINAL a times divisor := mul(11, 26)

Mobius inversion ORIGINAL inverse F(n/6) f(26) ORIGINAL a times divisor = 286

Mobius inversion ORIGINAL inverse F(n/6) f(26) plus ORIGINAL b := sum(286, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(26) plus ORIGINAL b = 287

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(12, 23, 144, 287)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 466

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 466)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 466

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

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

ORIGINAL positive integer absolute magnitude=12>1

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

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

upf extract divisor2 occurrence0 = 6

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

upf extract divisor2 occurrence1 = 3

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

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

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

upf canonical complete product factor2 = 12

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 := div(156, 12)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 = 13

Mobius inversion ORIGINAL inverse F(n/12):actual positive input=13;check complete integer range1..13

ALL positive divisors INCREASING=[1, 13];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor = 11

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

Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/12) f(13) ORIGINAL a times divisor := mul(11, 13)

Mobius inversion ORIGINAL inverse F(n/12) f(13) ORIGINAL a times divisor = 143

Mobius inversion ORIGINAL inverse F(n/12) f(13) plus ORIGINAL b := sum(143, 1)

Mobius inversion ORIGINAL inverse F(n/12) f(13) plus ORIGINAL b = 144

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum := sum(12, 144)

Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum = 156

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) := mul(0, 156)

Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:actual positive input for mu=13

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

ORIGINAL positive integer absolute magnitude=13>1

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

next divisor=2;previous remaining=13;all smaller divisors exhausted

extracted multiplicity=0;next remaining=13;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=13;all smaller divisors exhausted

extracted multiplicity=0;next remaining=13;remaining modulo3=1;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 13)

upf canonical complete product factor0 = 13

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=13

complete increasing prime multiset=[13];full product=13;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor13:input=13;COMPLETE prime multiset=[13];distinct primes=[13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 := div(156, 13)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 12

Mobius inversion ORIGINAL inverse F(n/13):actual positive input=12;check complete integer range1..12

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

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor := mul(11, 4)

Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor := mul(11, 6)

Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/13) f(12) ORIGINAL a times divisor := mul(11, 12)

Mobius inversion ORIGINAL inverse F(n/13) f(12) ORIGINAL a times divisor = 132

Mobius inversion ORIGINAL inverse F(n/13) f(12) plus ORIGINAL b := sum(132, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(12) plus ORIGINAL b = 133

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(12, 23, 34, 45, 67, 133)

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 314

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 314)

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -314

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor26:actual positive input for mu=26

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

ORIGINAL positive integer absolute magnitude=26>1

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

upf extract divisor2 occurrence0 := div(26, 2)

upf extract divisor2 occurrence0 = 13

next divisor=2;previous remaining=26;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=13;all smaller divisors exhausted

extracted multiplicity=0;next remaining=13;remaining modulo3=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, 13)

upf canonical complete product factor1 = 26

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=26

complete increasing prime multiset=[2, 13];full product=26;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor26:input=26;COMPLETE prime multiset=[2, 13];distinct primes=[2, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 := div(156, 26)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 6

Mobius inversion ORIGINAL inverse F(n/26):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/26) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/26) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/26) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/26) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/26) f(6) ORIGINAL a times divisor := mul(11, 6)

Mobius inversion ORIGINAL inverse F(n/26) f(6) ORIGINAL a times divisor = 66

Mobius inversion ORIGINAL inverse F(n/26) f(6) plus ORIGINAL b := sum(66, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(6) plus ORIGINAL b = 67

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum := sum(12, 23, 34, 67)

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 136

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) := mul(1, 136)

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 136

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:actual positive input for mu=39

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

ORIGINAL positive integer absolute magnitude=39>1

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

next divisor=2;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=0;next remaining=39;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(39, 3)

upf extract divisor3 occurrence0 = 13

next divisor=3;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;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, 13)

upf canonical complete product factor1 = 39

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=39

complete increasing prime multiset=[3, 13];full product=39;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:input=39;COMPLETE prime multiset=[3, 13];distinct primes=[3, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 := div(156, 39)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 = 4

Mobius inversion ORIGINAL inverse F(n/39):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 inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/39) f(4) ORIGINAL a times divisor := mul(11, 4)

Mobius inversion ORIGINAL inverse F(n/39) f(4) ORIGINAL a times divisor = 44

Mobius inversion ORIGINAL inverse F(n/39) f(4) plus ORIGINAL b := sum(44, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(4) plus ORIGINAL b = 45

Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum := sum(12, 23, 45)

Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum = 80

Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) := mul(1, 80)

Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) = 80

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor52:actual positive input for mu=52

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

ORIGINAL positive integer absolute magnitude=52>1

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

upf extract divisor2 occurrence0 := div(52, 2)

upf extract divisor2 occurrence0 = 26

upf extract divisor2 occurrence1 := div(26, 2)

upf extract divisor2 occurrence1 = 13

next divisor=2;previous remaining=52;all smaller divisors exhausted

extracted multiplicity=2;next remaining=13;remaining modulo2=1;extracted multiset=[2, 2]

next divisor=3;previous remaining=13;all smaller divisors exhausted

extracted multiplicity=0;next remaining=13;remaining modulo3=1;extracted multiset=[2, 2]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 13)

upf canonical complete product factor2 = 52

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=52

complete increasing prime multiset=[2, 2, 13];full product=52;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor52:input=52;COMPLETE prime multiset=[2, 2, 13];distinct primes=[2, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor52 := div(156, 52)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor52 = 3

Mobius inversion ORIGINAL inverse F(n/52):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/52) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/52) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/52) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/52) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/52) f(3) ORIGINAL a times divisor := mul(11, 3)

Mobius inversion ORIGINAL inverse F(n/52) f(3) ORIGINAL a times divisor = 33

Mobius inversion ORIGINAL inverse F(n/52) f(3) plus ORIGINAL b := sum(33, 1)

Mobius inversion ORIGINAL inverse F(n/52) f(3) plus ORIGINAL b = 34

Mobius inversion ORIGINAL inverse F(n/52) full forward divisor sum := sum(12, 34)

Mobius inversion ORIGINAL inverse F(n/52) full forward divisor sum = 46

Mobius inversion ORIGINAL inverse mu(52) times SAME F(n/52) := mul(0, 46)

Mobius inversion ORIGINAL inverse mu(52) times SAME F(n/52) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:actual positive input for mu=78

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

ORIGINAL positive integer absolute magnitude=78>1

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

upf extract divisor2 occurrence0 := div(78, 2)

upf extract divisor2 occurrence0 = 39

next divisor=2;previous remaining=78;all smaller divisors exhausted

extracted multiplicity=1;next remaining=39;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(39, 3)

upf extract divisor3 occurrence1 = 13

next divisor=3;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;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, 13)

upf canonical complete product factor2 = 78

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=78

complete increasing prime multiset=[2, 3, 13];full product=78;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:input=78;COMPLETE prime multiset=[2, 3, 13];distinct primes=[2, 3, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 := div(156, 78)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 = 2

Mobius inversion ORIGINAL inverse F(n/78):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/78) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/78) f(2) ORIGINAL a times divisor := mul(11, 2)

Mobius inversion ORIGINAL inverse F(n/78) f(2) ORIGINAL a times divisor = 22

Mobius inversion ORIGINAL inverse F(n/78) f(2) plus ORIGINAL b := sum(22, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(2) plus ORIGINAL b = 23

Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum := sum(12, 23)

Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum = 35

Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) := mul(-1, 35)

Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) = -35

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor156:actual positive input for mu=156

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

ORIGINAL positive integer absolute magnitude=156>1

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

upf extract divisor2 occurrence0 := div(156, 2)

upf extract divisor2 occurrence0 = 78

upf extract divisor2 occurrence1 := div(78, 2)

upf extract divisor2 occurrence1 = 39

next divisor=2;previous remaining=156;all smaller divisors exhausted

extracted multiplicity=2;next remaining=39;remaining modulo2=1;extracted multiset=[2, 2]

upf extract divisor3 occurrence2 := div(39, 3)

upf extract divisor3 occurrence2 = 13

next divisor=3;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo3=1;extracted multiset=[2, 2, 3]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

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

upf canonical complete product factor2 = 12

upf canonical complete product factor3 := mul(12, 13)

upf canonical complete product factor3 = 156

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=156

complete increasing prime multiset=[2, 2, 3, 13];full product=156;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor156:input=156;COMPLETE prime multiset=[2, 2, 3, 13];distinct primes=[2, 3, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor156 := div(156, 156)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor156 = 1

Mobius inversion ORIGINAL inverse F(n/156):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/156) f(1) ORIGINAL a times divisor := mul(11, 1)

Mobius inversion ORIGINAL inverse F(n/156) f(1) ORIGINAL a times divisor = 11

Mobius inversion ORIGINAL inverse F(n/156) f(1) plus ORIGINAL b := sum(11, 1)

Mobius inversion ORIGINAL inverse F(n/156) f(1) plus ORIGINAL b = 12

Mobius inversion ORIGINAL inverse F(n/156) full forward divisor sum := sum(12)

Mobius inversion ORIGINAL inverse F(n/156) full forward divisor sum = 12

Mobius inversion ORIGINAL inverse mu(156) times SAME F(n/156) := mul(0, 12)

Mobius inversion ORIGINAL inverse mu(156) times SAME F(n/156) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(4324, -1856, -1084, 0, 466, 0, -314, 136, 80, 0, -35, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 1717

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(11, 156)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 1716

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(1716, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 1717

Mobius inversion ORIGINAL inverse:actual n=156;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 156, 4324, 4324], [2, -1, 78, 1856, -1856], [3, -1, 52, 1084, -1084], [4, 0, 39, 620, 0], [6, 1, 26, 466, 466], [12, 0, 13, 156, 0], [13, -1, 12, 314, -314], [26, 1, 6, 136, 136], [39, 1, 4, 80, 80], [52, 0, 3, 46, 0], [78, -1, 2, 35, -35], [156, 0, 1, 12, 0]]

complete reconstructed=1717;direct f(n)=1717;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(-1856, -1084, 0, 466, 0, -314, 136, 80, 0, -35, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -2607

Mobius inversion complete minus partial := sub(1717, -2607)

Mobius inversion complete minus partial = 4324

ORIGINAL full divisors=[1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156]

complete=1717=f(n)=1717;partial=-2607;lost term=4324!=0;incomplete_certificate fixed No

Reverse checks

• S264 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S263 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S260 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(156) times SAME F(n/156) with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/156) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/156) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/156) f(1) ORIGINAL a times divisor 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor156 with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S250 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S249 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S244 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S242 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 inversion ORIGINAL inverse mu(78) times SAME F(n/78) with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S231 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 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 upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S224 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S222 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(52) times SAME F(n/52) with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/52) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/52) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/52) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/52) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S214 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/52) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor52 with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S209 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S204 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S194 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S188 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 inversion ORIGINAL inverse mu(26) times SAME F(n/26) with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(12) times SAME F(n/12) with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/12) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor12 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 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 upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(52) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(52) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) 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 inversion ORIGINAL inverse n divided by ORIGINAL divisor3 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 — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(78) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(78) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(156) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(156) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(52) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(52) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

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.