Question 22142: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=342, exact integer arithmetic function f(k)=a*k+b with a=17, 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=342, exact integer arithmetic function f(k)=a*k+b with a=17, 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=342;check complete integer range1..342

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342];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(342, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 342

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=342;check complete integer range1..342

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(34, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(51, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(17, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 102

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(102, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 103

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor := mul(17, 9)

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor = 153

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b := sum(153, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b = 154

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor := mul(17, 18)

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor = 306

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b := sum(306, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b = 307

Mobius inversion ORIGINAL inverse F(n/1) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/1) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/1) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/1) f(38) ORIGINAL a times divisor := mul(17, 38)

Mobius inversion ORIGINAL inverse F(n/1) f(38) ORIGINAL a times divisor = 646

Mobius inversion ORIGINAL inverse F(n/1) f(38) plus ORIGINAL b := sum(646, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(38) plus ORIGINAL b = 647

Mobius inversion ORIGINAL inverse F(n/1) f(57) ORIGINAL a times divisor := mul(17, 57)

Mobius inversion ORIGINAL inverse F(n/1) f(57) ORIGINAL a times divisor = 969

Mobius inversion ORIGINAL inverse F(n/1) f(57) plus ORIGINAL b := sum(969, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(57) plus ORIGINAL b = 970

Mobius inversion ORIGINAL inverse F(n/1) f(114) ORIGINAL a times divisor := mul(17, 114)

Mobius inversion ORIGINAL inverse F(n/1) f(114) ORIGINAL a times divisor = 1938

Mobius inversion ORIGINAL inverse F(n/1) f(114) plus ORIGINAL b := sum(1938, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(114) plus ORIGINAL b = 1939

Mobius inversion ORIGINAL inverse F(n/1) f(171) ORIGINAL a times divisor := mul(17, 171)

Mobius inversion ORIGINAL inverse F(n/1) f(171) ORIGINAL a times divisor = 2907

Mobius inversion ORIGINAL inverse F(n/1) f(171) plus ORIGINAL b := sum(2907, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(171) plus ORIGINAL b = 2908

Mobius inversion ORIGINAL inverse F(n/1) f(342) ORIGINAL a times divisor := mul(17, 342)

Mobius inversion ORIGINAL inverse F(n/1) f(342) ORIGINAL a times divisor = 5814

Mobius inversion ORIGINAL inverse F(n/1) f(342) plus ORIGINAL b := sum(5814, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(342) plus ORIGINAL b = 5815

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(18, 35, 52, 103, 154, 307, 324, 647, 970, 1939, 2908, 5815)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 13272

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 13272)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 13272

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(342, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 171

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=171;check complete integer range1..171

ALL positive divisors INCREASING=[1, 3, 9, 19, 57, 171];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor := mul(17, 9)

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor = 153

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b := sum(153, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b = 154

Mobius inversion ORIGINAL inverse F(n/2) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/2) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/2) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/2) f(57) ORIGINAL a times divisor := mul(17, 57)

Mobius inversion ORIGINAL inverse F(n/2) f(57) ORIGINAL a times divisor = 969

Mobius inversion ORIGINAL inverse F(n/2) f(57) plus ORIGINAL b := sum(969, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(57) plus ORIGINAL b = 970

Mobius inversion ORIGINAL inverse F(n/2) f(171) ORIGINAL a times divisor := mul(17, 171)

Mobius inversion ORIGINAL inverse F(n/2) f(171) ORIGINAL a times divisor = 2907

Mobius inversion ORIGINAL inverse F(n/2) f(171) plus ORIGINAL b := sum(2907, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(171) plus ORIGINAL b = 2908

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(18, 52, 154, 324, 970, 2908)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 4426

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 4426)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -4426

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(342, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 114

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=114;check complete integer range1..114

ALL positive divisors INCREASING=[1, 2, 3, 6, 19, 38, 57, 114];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(34, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 35

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b := sum(51, 1)

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

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor := mul(17, 6)

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor = 102

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b := sum(102, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b = 103

Mobius inversion ORIGINAL inverse F(n/3) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/3) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/3) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/3) f(38) ORIGINAL a times divisor := mul(17, 38)

Mobius inversion ORIGINAL inverse F(n/3) f(38) ORIGINAL a times divisor = 646

Mobius inversion ORIGINAL inverse F(n/3) f(38) plus ORIGINAL b := sum(646, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(38) plus ORIGINAL b = 647

Mobius inversion ORIGINAL inverse F(n/3) f(57) ORIGINAL a times divisor := mul(17, 57)

Mobius inversion ORIGINAL inverse F(n/3) f(57) ORIGINAL a times divisor = 969

Mobius inversion ORIGINAL inverse F(n/3) f(57) plus ORIGINAL b := sum(969, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(57) plus ORIGINAL b = 970

Mobius inversion ORIGINAL inverse F(n/3) f(114) ORIGINAL a times divisor := mul(17, 114)

Mobius inversion ORIGINAL inverse F(n/3) f(114) ORIGINAL a times divisor = 1938

Mobius inversion ORIGINAL inverse F(n/3) f(114) plus ORIGINAL b := sum(1938, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(114) plus ORIGINAL b = 1939

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(18, 35, 52, 103, 324, 647, 970, 1939)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 4088

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 4088)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -4088

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(342, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 57

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=57;check complete integer range1..57

ALL positive divisors INCREASING=[1, 3, 19, 57];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/6) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/6) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/6) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/6) f(57) ORIGINAL a times divisor := mul(17, 57)

Mobius inversion ORIGINAL inverse F(n/6) f(57) ORIGINAL a times divisor = 969

Mobius inversion ORIGINAL inverse F(n/6) f(57) plus ORIGINAL b := sum(969, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(57) plus ORIGINAL b = 970

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(18, 52, 324, 970)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 1364

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 1364)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 1364

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:actual positive input for mu=9

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

ORIGINAL positive integer absolute magnitude=9>1

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

next divisor=2;previous remaining=9;all smaller divisors exhausted

extracted multiplicity=0;next remaining=9;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(9, 3)

upf extract divisor3 occurrence0 = 3

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

upf extract divisor3 occurrence1 = 1

next divisor=3;previous remaining=9;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 9

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=9

complete increasing prime multiset=[3, 3];full product=9;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:input=9;COMPLETE prime multiset=[3, 3];distinct primes=[3]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 := div(342, 9)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 = 38

Mobius inversion ORIGINAL inverse F(n/9):actual positive input=38;check complete integer range1..38

ALL positive divisors INCREASING=[1, 2, 19, 38];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b := sum(34, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b = 35

Mobius inversion ORIGINAL inverse F(n/9) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/9) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/9) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/9) f(38) ORIGINAL a times divisor := mul(17, 38)

Mobius inversion ORIGINAL inverse F(n/9) f(38) ORIGINAL a times divisor = 646

Mobius inversion ORIGINAL inverse F(n/9) f(38) plus ORIGINAL b := sum(646, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(38) plus ORIGINAL b = 647

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum := sum(18, 35, 324, 647)

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum = 1024

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) := mul(0, 1024)

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:actual positive input for mu=18

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

ORIGINAL positive integer absolute magnitude=18>1

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

upf extract divisor2 occurrence0 := div(18, 2)

upf extract divisor2 occurrence0 = 9

next divisor=2;previous remaining=18;all smaller divisors exhausted

extracted multiplicity=1;next remaining=9;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(9, 3)

upf extract divisor3 occurrence1 = 3

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

upf extract divisor3 occurrence2 = 1

next divisor=3;previous remaining=9;all smaller divisors exhausted

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

upf canonical complete product factor2 = 18

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=18

complete increasing prime multiset=[2, 3, 3];full product=18;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:input=18;COMPLETE prime multiset=[2, 3, 3];distinct primes=[2, 3]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 := div(342, 18)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 = 19

Mobius inversion ORIGINAL inverse F(n/18):actual positive input=19;check complete integer range1..19

ALL positive divisors INCREASING=[1, 19];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/18) f(19) ORIGINAL a times divisor := mul(17, 19)

Mobius inversion ORIGINAL inverse F(n/18) f(19) ORIGINAL a times divisor = 323

Mobius inversion ORIGINAL inverse F(n/18) f(19) plus ORIGINAL b := sum(323, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(19) plus ORIGINAL b = 324

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum := sum(18, 324)

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum = 342

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) := mul(0, 342)

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor19:actual positive input for mu=19

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

ORIGINAL positive integer absolute magnitude=19>1

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

next divisor=2;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 19)

upf canonical complete product factor0 = 19

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=19

complete increasing prime multiset=[19];full product=19;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor19:input=19;COMPLETE prime multiset=[19];distinct primes=[19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 := div(342, 19)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 = 18

Mobius inversion ORIGINAL inverse F(n/19):actual positive input=18;check complete integer range1..18

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b := sum(34, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b = 35

Mobius inversion ORIGINAL inverse F(n/19) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/19) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/19) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/19) f(6) ORIGINAL a times divisor := mul(17, 6)

Mobius inversion ORIGINAL inverse F(n/19) f(6) ORIGINAL a times divisor = 102

Mobius inversion ORIGINAL inverse F(n/19) f(6) plus ORIGINAL b := sum(102, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(6) plus ORIGINAL b = 103

Mobius inversion ORIGINAL inverse F(n/19) f(9) ORIGINAL a times divisor := mul(17, 9)

Mobius inversion ORIGINAL inverse F(n/19) f(9) ORIGINAL a times divisor = 153

Mobius inversion ORIGINAL inverse F(n/19) f(9) plus ORIGINAL b := sum(153, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(9) plus ORIGINAL b = 154

Mobius inversion ORIGINAL inverse F(n/19) f(18) ORIGINAL a times divisor := mul(17, 18)

Mobius inversion ORIGINAL inverse F(n/19) f(18) ORIGINAL a times divisor = 306

Mobius inversion ORIGINAL inverse F(n/19) f(18) plus ORIGINAL b := sum(306, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(18) plus ORIGINAL b = 307

Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum := sum(18, 35, 52, 103, 154, 307)

Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum = 669

Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) := mul(-1, 669)

Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) = -669

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor38:actual positive input for mu=38

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

ORIGINAL positive integer absolute magnitude=38>1

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

upf extract divisor2 occurrence0 := div(38, 2)

upf extract divisor2 occurrence0 = 19

next divisor=2;previous remaining=38;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 19)

upf canonical complete product factor1 = 38

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=38

complete increasing prime multiset=[2, 19];full product=38;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor38:input=38;COMPLETE prime multiset=[2, 19];distinct primes=[2, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 := div(342, 38)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 = 9

Mobius inversion ORIGINAL inverse F(n/38):actual positive input=9;check complete integer range1..9

ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/38) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/38) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/38) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/38) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/38) f(9) ORIGINAL a times divisor := mul(17, 9)

Mobius inversion ORIGINAL inverse F(n/38) f(9) ORIGINAL a times divisor = 153

Mobius inversion ORIGINAL inverse F(n/38) f(9) plus ORIGINAL b := sum(153, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(9) plus ORIGINAL b = 154

Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum := sum(18, 52, 154)

Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum = 224

Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) := mul(1, 224)

Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) = 224

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor57:actual positive input for mu=57

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

ORIGINAL positive integer absolute magnitude=57>1

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

next divisor=2;previous remaining=57;all smaller divisors exhausted

extracted multiplicity=0;next remaining=57;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(57, 3)

upf extract divisor3 occurrence0 = 19

next divisor=3;previous remaining=57;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo3=1;extracted multiset=[3]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[3]

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

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 19)

upf canonical complete product factor1 = 57

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=57

complete increasing prime multiset=[3, 19];full product=57;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor57:input=57;COMPLETE prime multiset=[3, 19];distinct primes=[3, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor57 := div(342, 57)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor57 = 6

Mobius inversion ORIGINAL inverse F(n/57):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/57) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/57) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/57) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/57) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/57) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/57) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/57) f(2) plus ORIGINAL b := sum(34, 1)

Mobius inversion ORIGINAL inverse F(n/57) f(2) plus ORIGINAL b = 35

Mobius inversion ORIGINAL inverse F(n/57) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/57) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/57) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/57) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/57) f(6) ORIGINAL a times divisor := mul(17, 6)

Mobius inversion ORIGINAL inverse F(n/57) f(6) ORIGINAL a times divisor = 102

Mobius inversion ORIGINAL inverse F(n/57) f(6) plus ORIGINAL b := sum(102, 1)

Mobius inversion ORIGINAL inverse F(n/57) f(6) plus ORIGINAL b = 103

Mobius inversion ORIGINAL inverse F(n/57) full forward divisor sum := sum(18, 35, 52, 103)

Mobius inversion ORIGINAL inverse F(n/57) full forward divisor sum = 208

Mobius inversion ORIGINAL inverse mu(57) times SAME F(n/57) := mul(1, 208)

Mobius inversion ORIGINAL inverse mu(57) times SAME F(n/57) = 208

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor114:actual positive input for mu=114

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

ORIGINAL positive integer absolute magnitude=114>1

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

upf extract divisor2 occurrence0 := div(114, 2)

upf extract divisor2 occurrence0 = 57

next divisor=2;previous remaining=114;all smaller divisors exhausted

extracted multiplicity=1;next remaining=57;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(57, 3)

upf extract divisor3 occurrence1 = 19

next divisor=3;previous remaining=57;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo3=1;extracted multiset=[2, 3]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;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, 19)

upf canonical complete product factor2 = 114

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=114

complete increasing prime multiset=[2, 3, 19];full product=114;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor114:input=114;COMPLETE prime multiset=[2, 3, 19];distinct primes=[2, 3, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor114 := div(342, 114)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor114 = 3

Mobius inversion ORIGINAL inverse F(n/114):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/114) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/114) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/114) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/114) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/114) f(3) ORIGINAL a times divisor := mul(17, 3)

Mobius inversion ORIGINAL inverse F(n/114) f(3) ORIGINAL a times divisor = 51

Mobius inversion ORIGINAL inverse F(n/114) f(3) plus ORIGINAL b := sum(51, 1)

Mobius inversion ORIGINAL inverse F(n/114) f(3) plus ORIGINAL b = 52

Mobius inversion ORIGINAL inverse F(n/114) full forward divisor sum := sum(18, 52)

Mobius inversion ORIGINAL inverse F(n/114) full forward divisor sum = 70

Mobius inversion ORIGINAL inverse mu(114) times SAME F(n/114) := mul(-1, 70)

Mobius inversion ORIGINAL inverse mu(114) times SAME F(n/114) = -70

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor171:actual positive input for mu=171

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

ORIGINAL positive integer absolute magnitude=171>1

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

next divisor=2;previous remaining=171;all smaller divisors exhausted

extracted multiplicity=0;next remaining=171;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(171, 3)

upf extract divisor3 occurrence0 = 57

upf extract divisor3 occurrence1 := div(57, 3)

upf extract divisor3 occurrence1 = 19

next divisor=3;previous remaining=171;all smaller divisors exhausted

extracted multiplicity=2;next remaining=19;remaining modulo3=1;extracted multiset=[3, 3]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[3, 3]

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 9

upf canonical complete product factor2 := mul(9, 19)

upf canonical complete product factor2 = 171

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=171

complete increasing prime multiset=[3, 3, 19];full product=171;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor171:input=171;COMPLETE prime multiset=[3, 3, 19];distinct primes=[3, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor171 := div(342, 171)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor171 = 2

Mobius inversion ORIGINAL inverse F(n/171):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/171) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/171) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/171) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/171) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/171) f(2) ORIGINAL a times divisor := mul(17, 2)

Mobius inversion ORIGINAL inverse F(n/171) f(2) ORIGINAL a times divisor = 34

Mobius inversion ORIGINAL inverse F(n/171) f(2) plus ORIGINAL b := sum(34, 1)

Mobius inversion ORIGINAL inverse F(n/171) f(2) plus ORIGINAL b = 35

Mobius inversion ORIGINAL inverse F(n/171) full forward divisor sum := sum(18, 35)

Mobius inversion ORIGINAL inverse F(n/171) full forward divisor sum = 53

Mobius inversion ORIGINAL inverse mu(171) times SAME F(n/171) := mul(0, 53)

Mobius inversion ORIGINAL inverse mu(171) times SAME F(n/171) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor342:actual positive input for mu=342

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

ORIGINAL positive integer absolute magnitude=342>1

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

upf extract divisor2 occurrence0 := div(342, 2)

upf extract divisor2 occurrence0 = 171

next divisor=2;previous remaining=342;all smaller divisors exhausted

extracted multiplicity=1;next remaining=171;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(171, 3)

upf extract divisor3 occurrence1 = 57

upf extract divisor3 occurrence2 := div(57, 3)

upf extract divisor3 occurrence2 = 19

next divisor=3;previous remaining=171;all smaller divisors exhausted

extracted multiplicity=2;next remaining=19;remaining modulo3=1;extracted multiset=[2, 3, 3]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[2, 3, 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, 3)

upf canonical complete product factor2 = 18

upf canonical complete product factor3 := mul(18, 19)

upf canonical complete product factor3 = 342

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=342

complete increasing prime multiset=[2, 3, 3, 19];full product=342;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor342:input=342;COMPLETE prime multiset=[2, 3, 3, 19];distinct primes=[2, 3, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor342 := div(342, 342)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor342 = 1

Mobius inversion ORIGINAL inverse F(n/342):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/342) f(1) ORIGINAL a times divisor := mul(17, 1)

Mobius inversion ORIGINAL inverse F(n/342) f(1) ORIGINAL a times divisor = 17

Mobius inversion ORIGINAL inverse F(n/342) f(1) plus ORIGINAL b := sum(17, 1)

Mobius inversion ORIGINAL inverse F(n/342) f(1) plus ORIGINAL b = 18

Mobius inversion ORIGINAL inverse F(n/342) full forward divisor sum := sum(18)

Mobius inversion ORIGINAL inverse F(n/342) full forward divisor sum = 18

Mobius inversion ORIGINAL inverse mu(342) times SAME F(n/342) := mul(0, 18)

Mobius inversion ORIGINAL inverse mu(342) times SAME F(n/342) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(13272, -4426, -4088, 1364, 0, 0, -669, 224, 208, -70, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 5815

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(17, 342)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 5814

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(5814, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 5815

Mobius inversion ORIGINAL inverse:actual n=342;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 342, 13272, 13272], [2, -1, 171, 4426, -4426], [3, -1, 114, 4088, -4088], [6, 1, 57, 1364, 1364], [9, 0, 38, 1024, 0], [18, 0, 19, 342, 0], [19, -1, 18, 669, -669], [38, 1, 9, 224, 224], [57, 1, 6, 208, 208], [114, -1, 3, 70, -70], [171, 0, 2, 53, 0], [342, 0, 1, 18, 0]]

complete reconstructed=5815;direct f(n)=5815;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(-4426, -4088, 1364, 0, 0, -669, 224, 208, -70, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -7457

Mobius inversion complete minus partial := sub(5815, -7457)

Mobius inversion complete minus partial = 13272

ORIGINAL full divisors=[1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342]

complete=5815=f(n)=5815;partial=-7457;lost term=13272!=0;incomplete_certificate fixed No

Reverse checks

• 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 Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S269 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S266 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(342) times SAME F(n/342) with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/342) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/342) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/342) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S261 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor342 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S258 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• 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 extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S246 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(171) times SAME F(n/171) with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/171) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/171) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/171) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/171) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/171) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor171 with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S231 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S226 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(114) times SAME F(n/114) with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/114) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/114) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/114) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/114) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/114) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 n divided by ORIGINAL divisor114 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S216 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• 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 upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• 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 extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(57) times SAME F(n/57) with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/57) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor57 with exact rational arithmetic, not rounded decimals.
• S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S192 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S184 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S172 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 — 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 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 occurrence0 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(38) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute upf extract divisor3 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(57) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(57) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S089 · 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.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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(114) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(114) 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(57) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(57) 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(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(38) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(19) 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(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(6) 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(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S063 · 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.
• S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S061 · 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.
• 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 inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(171) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(171) 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(57) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(57) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(19) 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(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(9) 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(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S042 · 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.
• 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(342) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(342) 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(171) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(171) 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(114) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(114) 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(57) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(57) 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(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(38) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(19) 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(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(18) 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(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(9) 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(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S013 · 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.
• 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

(342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, 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.