Question 22141: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=306, exact integer arithmetic function f(k)=a*k+b with a=16, 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=306, exact integer arithmetic function f(k)=a*k+b with a=16, 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=306;check complete integer range1..306

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306];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(306, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 306

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=306;check complete integer range1..306

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306];not only primes/proper divisors or all integers<=n

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

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(32, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(16, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 96

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(96, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 97

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor := mul(16, 9)

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/1) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/1) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/1) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor := mul(16, 18)

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor = 288

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b := sum(288, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b = 289

Mobius inversion ORIGINAL inverse F(n/1) f(34) ORIGINAL a times divisor := mul(16, 34)

Mobius inversion ORIGINAL inverse F(n/1) f(34) ORIGINAL a times divisor = 544

Mobius inversion ORIGINAL inverse F(n/1) f(34) plus ORIGINAL b := sum(544, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(34) plus ORIGINAL b = 545

Mobius inversion ORIGINAL inverse F(n/1) f(51) ORIGINAL a times divisor := mul(16, 51)

Mobius inversion ORIGINAL inverse F(n/1) f(51) ORIGINAL a times divisor = 816

Mobius inversion ORIGINAL inverse F(n/1) f(51) plus ORIGINAL b := sum(816, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(51) plus ORIGINAL b = 817

Mobius inversion ORIGINAL inverse F(n/1) f(102) ORIGINAL a times divisor := mul(16, 102)

Mobius inversion ORIGINAL inverse F(n/1) f(102) ORIGINAL a times divisor = 1632

Mobius inversion ORIGINAL inverse F(n/1) f(102) plus ORIGINAL b := sum(1632, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(102) plus ORIGINAL b = 1633

Mobius inversion ORIGINAL inverse F(n/1) f(153) ORIGINAL a times divisor := mul(16, 153)

Mobius inversion ORIGINAL inverse F(n/1) f(153) ORIGINAL a times divisor = 2448

Mobius inversion ORIGINAL inverse F(n/1) f(153) plus ORIGINAL b := sum(2448, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(153) plus ORIGINAL b = 2449

Mobius inversion ORIGINAL inverse F(n/1) f(306) ORIGINAL a times divisor := mul(16, 306)

Mobius inversion ORIGINAL inverse F(n/1) f(306) ORIGINAL a times divisor = 4896

Mobius inversion ORIGINAL inverse F(n/1) f(306) plus ORIGINAL b := sum(4896, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(306) plus ORIGINAL b = 4897

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(17, 33, 49, 97, 145, 273, 289, 545, 817, 1633, 2449, 4897)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 11244

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 11244)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 11244

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(306, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 153

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=153;check complete integer range1..153

ALL positive divisors INCREASING=[1, 3, 9, 17, 51, 153];not only primes/proper divisors or all integers<=n

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor := mul(16, 9)

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/2) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/2) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/2) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/2) f(51) ORIGINAL a times divisor := mul(16, 51)

Mobius inversion ORIGINAL inverse F(n/2) f(51) ORIGINAL a times divisor = 816

Mobius inversion ORIGINAL inverse F(n/2) f(51) plus ORIGINAL b := sum(816, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(51) plus ORIGINAL b = 817

Mobius inversion ORIGINAL inverse F(n/2) f(153) ORIGINAL a times divisor := mul(16, 153)

Mobius inversion ORIGINAL inverse F(n/2) f(153) ORIGINAL a times divisor = 2448

Mobius inversion ORIGINAL inverse F(n/2) f(153) plus ORIGINAL b := sum(2448, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(153) plus ORIGINAL b = 2449

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(17, 49, 145, 273, 817, 2449)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 3750

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 3750)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -3750

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(306, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 102

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=102;check complete integer range1..102

ALL positive divisors INCREASING=[1, 2, 3, 6, 17, 34, 51, 102];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 16

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

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

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(16, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(32, 1)

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

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor := mul(16, 6)

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor = 96

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b := sum(96, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b = 97

Mobius inversion ORIGINAL inverse F(n/3) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/3) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/3) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/3) f(34) ORIGINAL a times divisor := mul(16, 34)

Mobius inversion ORIGINAL inverse F(n/3) f(34) ORIGINAL a times divisor = 544

Mobius inversion ORIGINAL inverse F(n/3) f(34) plus ORIGINAL b := sum(544, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(34) plus ORIGINAL b = 545

Mobius inversion ORIGINAL inverse F(n/3) f(51) ORIGINAL a times divisor := mul(16, 51)

Mobius inversion ORIGINAL inverse F(n/3) f(51) ORIGINAL a times divisor = 816

Mobius inversion ORIGINAL inverse F(n/3) f(51) plus ORIGINAL b := sum(816, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(51) plus ORIGINAL b = 817

Mobius inversion ORIGINAL inverse F(n/3) f(102) ORIGINAL a times divisor := mul(16, 102)

Mobius inversion ORIGINAL inverse F(n/3) f(102) ORIGINAL a times divisor = 1632

Mobius inversion ORIGINAL inverse F(n/3) f(102) plus ORIGINAL b := sum(1632, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(102) plus ORIGINAL b = 1633

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(17, 33, 49, 97, 273, 545, 817, 1633)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 3464

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 3464)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -3464

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(306, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 51

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=51;check complete integer range1..51

ALL positive divisors INCREASING=[1, 3, 17, 51];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(16, 1)

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

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/6) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/6) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/6) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/6) f(51) ORIGINAL a times divisor := mul(16, 51)

Mobius inversion ORIGINAL inverse F(n/6) f(51) ORIGINAL a times divisor = 816

Mobius inversion ORIGINAL inverse F(n/6) f(51) plus ORIGINAL b := sum(816, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(51) plus ORIGINAL b = 817

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(17, 49, 273, 817)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 1156

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 1156)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 1156

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(306, 9)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 = 34

Mobius inversion ORIGINAL inverse F(n/9):actual positive input=34;check complete integer range1..34

ALL positive divisors INCREASING=[1, 2, 17, 34];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b := sum(16, 1)

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

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor := mul(16, 2)

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b := sum(32, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b = 33

Mobius inversion ORIGINAL inverse F(n/9) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/9) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/9) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/9) f(34) ORIGINAL a times divisor := mul(16, 34)

Mobius inversion ORIGINAL inverse F(n/9) f(34) ORIGINAL a times divisor = 544

Mobius inversion ORIGINAL inverse F(n/9) f(34) plus ORIGINAL b := sum(544, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(34) plus ORIGINAL b = 545

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum := sum(17, 33, 273, 545)

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum = 868

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) := mul(0, 868)

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

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor17:actual positive input for mu=17

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

ORIGINAL positive integer absolute magnitude=17>1

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

next divisor=2;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=1;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 17)

upf canonical complete product factor0 = 17

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=17

complete increasing prime multiset=[17];full product=17;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor17:input=17;COMPLETE prime multiset=[17];distinct primes=[17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor17 := div(306, 17)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor17 = 18

Mobius inversion ORIGINAL inverse F(n/17):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/17) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/17) f(1) plus ORIGINAL b := sum(16, 1)

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

Mobius inversion ORIGINAL inverse F(n/17) f(2) ORIGINAL a times divisor := mul(16, 2)

Mobius inversion ORIGINAL inverse F(n/17) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/17) f(2) plus ORIGINAL b := sum(32, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(2) plus ORIGINAL b = 33

Mobius inversion ORIGINAL inverse F(n/17) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/17) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/17) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/17) f(6) ORIGINAL a times divisor := mul(16, 6)

Mobius inversion ORIGINAL inverse F(n/17) f(6) ORIGINAL a times divisor = 96

Mobius inversion ORIGINAL inverse F(n/17) f(6) plus ORIGINAL b := sum(96, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(6) plus ORIGINAL b = 97

Mobius inversion ORIGINAL inverse F(n/17) f(9) ORIGINAL a times divisor := mul(16, 9)

Mobius inversion ORIGINAL inverse F(n/17) f(9) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/17) f(9) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(9) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/17) f(18) ORIGINAL a times divisor := mul(16, 18)

Mobius inversion ORIGINAL inverse F(n/17) f(18) ORIGINAL a times divisor = 288

Mobius inversion ORIGINAL inverse F(n/17) f(18) plus ORIGINAL b := sum(288, 1)

Mobius inversion ORIGINAL inverse F(n/17) f(18) plus ORIGINAL b = 289

Mobius inversion ORIGINAL inverse F(n/17) full forward divisor sum := sum(17, 33, 49, 97, 145, 289)

Mobius inversion ORIGINAL inverse F(n/17) full forward divisor sum = 630

Mobius inversion ORIGINAL inverse mu(17) times SAME F(n/17) := mul(-1, 630)

Mobius inversion ORIGINAL inverse mu(17) times SAME F(n/17) = -630

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(306, 18)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 = 17

Mobius inversion ORIGINAL inverse F(n/18):actual positive input=17;check complete integer range1..17

ALL positive divisors INCREASING=[1, 17];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b := sum(16, 1)

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

Mobius inversion ORIGINAL inverse F(n/18) f(17) ORIGINAL a times divisor := mul(16, 17)

Mobius inversion ORIGINAL inverse F(n/18) f(17) ORIGINAL a times divisor = 272

Mobius inversion ORIGINAL inverse F(n/18) f(17) plus ORIGINAL b := sum(272, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(17) plus ORIGINAL b = 273

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum := sum(17, 273)

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum = 290

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) := mul(0, 290)

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

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor34:actual positive input for mu=34

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

ORIGINAL positive integer absolute magnitude=34>1

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

upf extract divisor2 occurrence0 := div(34, 2)

upf extract divisor2 occurrence0 = 17

next divisor=2;previous remaining=34;all smaller divisors exhausted

extracted multiplicity=1;next remaining=17;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=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, 17)

upf canonical complete product factor1 = 34

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=34

complete increasing prime multiset=[2, 17];full product=34;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor34:input=34;COMPLETE prime multiset=[2, 17];distinct primes=[2, 17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor34 := div(306, 34)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor34 = 9

Mobius inversion ORIGINAL inverse F(n/34):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/34) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/34) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/34) f(1) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/34) f(1) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/34) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/34) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/34) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/34) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/34) f(9) ORIGINAL a times divisor := mul(16, 9)

Mobius inversion ORIGINAL inverse F(n/34) f(9) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/34) f(9) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/34) f(9) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/34) full forward divisor sum := sum(17, 49, 145)

Mobius inversion ORIGINAL inverse F(n/34) full forward divisor sum = 211

Mobius inversion ORIGINAL inverse mu(34) times SAME F(n/34) := mul(1, 211)

Mobius inversion ORIGINAL inverse mu(34) times SAME F(n/34) = 211

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor51:actual positive input for mu=51

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

ORIGINAL positive integer absolute magnitude=51>1

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

next divisor=2;previous remaining=51;all smaller divisors exhausted

extracted multiplicity=0;next remaining=51;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(51, 3)

upf extract divisor3 occurrence0 = 17

next divisor=3;previous remaining=51;all smaller divisors exhausted

extracted multiplicity=1;next remaining=17;remaining modulo3=2;extracted multiset=[3]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=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, 17)

upf canonical complete product factor1 = 51

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=51

complete increasing prime multiset=[3, 17];full product=51;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor51:input=51;COMPLETE prime multiset=[3, 17];distinct primes=[3, 17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor51 := div(306, 51)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor51 = 6

Mobius inversion ORIGINAL inverse F(n/51):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/51) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/51) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/51) f(1) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/51) f(1) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/51) f(2) ORIGINAL a times divisor := mul(16, 2)

Mobius inversion ORIGINAL inverse F(n/51) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/51) f(2) plus ORIGINAL b := sum(32, 1)

Mobius inversion ORIGINAL inverse F(n/51) f(2) plus ORIGINAL b = 33

Mobius inversion ORIGINAL inverse F(n/51) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/51) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/51) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/51) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/51) f(6) ORIGINAL a times divisor := mul(16, 6)

Mobius inversion ORIGINAL inverse F(n/51) f(6) ORIGINAL a times divisor = 96

Mobius inversion ORIGINAL inverse F(n/51) f(6) plus ORIGINAL b := sum(96, 1)

Mobius inversion ORIGINAL inverse F(n/51) f(6) plus ORIGINAL b = 97

Mobius inversion ORIGINAL inverse F(n/51) full forward divisor sum := sum(17, 33, 49, 97)

Mobius inversion ORIGINAL inverse F(n/51) full forward divisor sum = 196

Mobius inversion ORIGINAL inverse mu(51) times SAME F(n/51) := mul(1, 196)

Mobius inversion ORIGINAL inverse mu(51) times SAME F(n/51) = 196

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor102:actual positive input for mu=102

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

ORIGINAL positive integer absolute magnitude=102>1

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

upf extract divisor2 occurrence0 := div(102, 2)

upf extract divisor2 occurrence0 = 51

next divisor=2;previous remaining=102;all smaller divisors exhausted

extracted multiplicity=1;next remaining=51;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(51, 3)

upf extract divisor3 occurrence1 = 17

next divisor=3;previous remaining=51;all smaller divisors exhausted

extracted multiplicity=1;next remaining=17;remaining modulo3=2;extracted multiset=[2, 3]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=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, 17)

upf canonical complete product factor2 = 102

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=102

complete increasing prime multiset=[2, 3, 17];full product=102;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor102:input=102;COMPLETE prime multiset=[2, 3, 17];distinct primes=[2, 3, 17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor102 := div(306, 102)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor102 = 3

Mobius inversion ORIGINAL inverse F(n/102):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/102) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/102) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/102) f(1) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/102) f(1) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/102) f(3) ORIGINAL a times divisor := mul(16, 3)

Mobius inversion ORIGINAL inverse F(n/102) f(3) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/102) f(3) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/102) f(3) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/102) full forward divisor sum := sum(17, 49)

Mobius inversion ORIGINAL inverse F(n/102) full forward divisor sum = 66

Mobius inversion ORIGINAL inverse mu(102) times SAME F(n/102) := mul(-1, 66)

Mobius inversion ORIGINAL inverse mu(102) times SAME F(n/102) = -66

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor153:actual positive input for mu=153

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

ORIGINAL positive integer absolute magnitude=153>1

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

next divisor=2;previous remaining=153;all smaller divisors exhausted

extracted multiplicity=0;next remaining=153;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(153, 3)

upf extract divisor3 occurrence0 = 51

upf extract divisor3 occurrence1 := div(51, 3)

upf extract divisor3 occurrence1 = 17

next divisor=3;previous remaining=153;all smaller divisors exhausted

extracted multiplicity=2;next remaining=17;remaining modulo3=2;extracted multiset=[3, 3]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=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

upf canonical complete product factor2 := mul(9, 17)

upf canonical complete product factor2 = 153

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=153

complete increasing prime multiset=[3, 3, 17];full product=153;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor153:input=153;COMPLETE prime multiset=[3, 3, 17];distinct primes=[3, 17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor153 := div(306, 153)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor153 = 2

Mobius inversion ORIGINAL inverse F(n/153):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/153) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/153) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/153) f(1) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/153) f(1) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/153) f(2) ORIGINAL a times divisor := mul(16, 2)

Mobius inversion ORIGINAL inverse F(n/153) f(2) ORIGINAL a times divisor = 32

Mobius inversion ORIGINAL inverse F(n/153) f(2) plus ORIGINAL b := sum(32, 1)

Mobius inversion ORIGINAL inverse F(n/153) f(2) plus ORIGINAL b = 33

Mobius inversion ORIGINAL inverse F(n/153) full forward divisor sum := sum(17, 33)

Mobius inversion ORIGINAL inverse F(n/153) full forward divisor sum = 50

Mobius inversion ORIGINAL inverse mu(153) times SAME F(n/153) := mul(0, 50)

Mobius inversion ORIGINAL inverse mu(153) times SAME F(n/153) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor306:actual positive input for mu=306

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

ORIGINAL positive integer absolute magnitude=306>1

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

upf extract divisor2 occurrence0 := div(306, 2)

upf extract divisor2 occurrence0 = 153

next divisor=2;previous remaining=306;all smaller divisors exhausted

extracted multiplicity=1;next remaining=153;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(153, 3)

upf extract divisor3 occurrence1 = 51

upf extract divisor3 occurrence2 := div(51, 3)

upf extract divisor3 occurrence2 = 17

next divisor=3;previous remaining=153;all smaller divisors exhausted

extracted multiplicity=2;next remaining=17;remaining modulo3=2;extracted multiset=[2, 3, 3]

next divisor=4;previous remaining=17;all smaller divisors exhausted

extracted multiplicity=0;next remaining=17;remaining modulo4=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

upf canonical complete product factor3 := mul(18, 17)

upf canonical complete product factor3 = 306

exhausted trial divisors below5;terminal remaining=17;d*d>remaining;ORIGINAL n=306

complete increasing prime multiset=[2, 3, 3, 17];full product=306;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor306:input=306;COMPLETE prime multiset=[2, 3, 3, 17];distinct primes=[2, 3, 17]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor306 := div(306, 306)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor306 = 1

Mobius inversion ORIGINAL inverse F(n/306):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/306) f(1) ORIGINAL a times divisor := mul(16, 1)

Mobius inversion ORIGINAL inverse F(n/306) f(1) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/306) f(1) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/306) f(1) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/306) full forward divisor sum := sum(17)

Mobius inversion ORIGINAL inverse F(n/306) full forward divisor sum = 17

Mobius inversion ORIGINAL inverse mu(306) times SAME F(n/306) := mul(0, 17)

Mobius inversion ORIGINAL inverse mu(306) times SAME F(n/306) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(11244, -3750, -3464, 1156, 0, -630, 0, 211, 196, -66, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 4897

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(16, 306)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 4896

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(4896, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 4897

Mobius inversion ORIGINAL inverse:actual n=306;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 306, 11244, 11244], [2, -1, 153, 3750, -3750], [3, -1, 102, 3464, -3464], [6, 1, 51, 1156, 1156], [9, 0, 34, 868, 0], [17, -1, 18, 630, -630], [18, 0, 17, 290, 0], [34, 1, 9, 211, 211], [51, 1, 6, 196, 196], [102, -1, 3, 66, -66], [153, 0, 2, 50, 0], [306, 0, 1, 17, 0]]

complete reconstructed=4897;direct f(n)=4897;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(-3750, -3464, 1156, 0, -630, 0, 211, 196, -66, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -6347

Mobius inversion complete minus partial := sub(4897, -6347)

Mobius inversion complete minus partial = 11244

ORIGINAL full divisors=[1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306]

complete=4897=f(n)=4897;partial=-6347;lost term=11244!=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(306) times SAME F(n/306) with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/306) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/306) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/306) 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 divisor306 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(153) times SAME F(n/153) with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/153) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/153) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/153) 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/153) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/153) 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 divisor153 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(102) times SAME F(n/102) with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/102) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/102) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/102) 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/102) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/102) 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 divisor102 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(51) times SAME F(n/51) with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) 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/51) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) 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/51) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) 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/51) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/51) 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 divisor51 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(34) times SAME F(n/34) with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/34) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/34) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/34) 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/34) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/34) 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/34) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/34) 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 divisor34 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(18) times SAME F(n/18) with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(17) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S159 · 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.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 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 — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(17) times SAME F(n/17) with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/17) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor17 with exact rational arithmetic, not rounded decimals.
• 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 — 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 canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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(34) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(34) 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(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(17) 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(51) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(51) 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(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(17) 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(102) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(102) 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(51) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(51) 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(34) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(34) 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(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(17) 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(153) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(153) 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(51) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(51) 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(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(17) 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(306) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(306) 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(153) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(153) 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(102) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(102) 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(51) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(51) 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(34) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(34) 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(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · 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.
• S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(17) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(17) 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

(306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, 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.