Question 22145: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=462, exact integer arithmetic function f(k)=a*k+b with a=20, 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=462, exact integer arithmetic function f(k)=a*k+b with a=20, 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=462;check complete integer range1..462

ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];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(462, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 462

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=462;check complete integer range1..462

ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(20, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(20, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor := mul(20, 14)

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 280

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b := sum(280, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 281

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor := mul(20, 21)

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 420

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b := sum(420, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 421

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(20, 22)

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 440

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(440, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 441

Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor := mul(20, 33)

Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor = 660

Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b := sum(660, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b = 661

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor := mul(20, 42)

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 840

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b := sum(840, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 841

Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor := mul(20, 66)

Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor = 1320

Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b := sum(1320, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b = 1321

Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor := mul(20, 77)

Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor = 1540

Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b := sum(1540, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b = 1541

Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor := mul(20, 154)

Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor = 3080

Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b := sum(3080, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b = 3081

Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor := mul(20, 231)

Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor = 4620

Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b := sum(4620, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b = 4621

Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor := mul(20, 462)

Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor = 9240

Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b := sum(9240, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b = 9241

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(21, 41, 61, 121, 141, 221, 281, 421, 441, 661, 841, 1321, 1541, 3081, 4621, 9241)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 23056

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 23056)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 23056

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(462, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 231

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=231;check complete integer range1..231

ALL positive divisors INCREASING=[1, 3, 7, 11, 21, 33, 77, 231];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(20, 1)

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

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor := mul(20, 21)

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 420

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b := sum(420, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 421

Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor := mul(20, 33)

Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor = 660

Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b := sum(660, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b = 661

Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor := mul(20, 77)

Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor = 1540

Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b := sum(1540, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b = 1541

Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor := mul(20, 231)

Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor = 4620

Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b := sum(4620, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b = 4621

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(21, 61, 141, 221, 421, 661, 1541, 4621)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 7688

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 7688)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -7688

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(462, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 154

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=154;check complete integer range1..154

ALL positive divisors INCREASING=[1, 2, 7, 11, 14, 22, 77, 154];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(20, 14)

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 280

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b := sum(280, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 281

Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor := mul(20, 22)

Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor = 440

Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b := sum(440, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b = 441

Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor := mul(20, 77)

Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor = 1540

Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b := sum(1540, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b = 1541

Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor := mul(20, 154)

Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor = 3080

Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b := sum(3080, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b = 3081

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(21, 41, 141, 221, 281, 441, 1541, 3081)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 5768

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 5768)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -5768

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(462, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 77

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=77;check complete integer range1..77

ALL positive divisors INCREASING=[1, 7, 11, 77];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(20, 1)

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

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor := mul(20, 77)

Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor = 1540

Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b := sum(1540, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b = 1541

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(21, 141, 221, 1541)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 1924

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 1924)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 1924

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

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

ORIGINAL positive integer absolute magnitude=7>1

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

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

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

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

upf canonical complete product factor0 = 7

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(462, 7)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 66

Mobius inversion ORIGINAL inverse F(n/7):actual positive input=66;check complete integer range1..66

ALL positive divisors INCREASING=[1, 2, 3, 6, 11, 22, 33, 66];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor := mul(20, 6)

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor := mul(20, 22)

Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor = 440

Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b := sum(440, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b = 441

Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor := mul(20, 33)

Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor = 660

Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b := sum(660, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b = 661

Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor := mul(20, 66)

Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor = 1320

Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b := sum(1320, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b = 1321

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(21, 41, 61, 121, 221, 441, 661, 1321)

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 2888

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 2888)

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -2888

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:actual positive input for mu=11

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

ORIGINAL positive integer absolute magnitude=11>1

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

next divisor=2;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 11)

upf canonical complete product factor0 = 11

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=11

complete increasing prime multiset=[11];full product=11;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:input=11;COMPLETE prime multiset=[11];distinct primes=[11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 := div(462, 11)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 42

Mobius inversion ORIGINAL inverse F(n/11):actual positive input=42;check complete integer range1..42

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

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor := mul(20, 6)

Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor := mul(20, 14)

Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor = 280

Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b := sum(280, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b = 281

Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor := mul(20, 21)

Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor = 420

Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b := sum(420, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b = 421

Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor := mul(20, 42)

Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor = 840

Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b := sum(840, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b = 841

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(21, 41, 61, 121, 141, 281, 421, 841)

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 1928

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 1928)

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -1928

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

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

ORIGINAL positive integer absolute magnitude=14>1

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

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

upf extract divisor2 occurrence0 = 7

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

extracted multiplicity=1;next remaining=7;remaining modulo2=1;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 7)

upf canonical complete product factor1 = 14

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(462, 14)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 33

Mobius inversion ORIGINAL inverse F(n/14):actual positive input=33;check complete integer range1..33

ALL positive divisors INCREASING=[1, 3, 11, 33];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor := mul(20, 33)

Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor = 660

Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b := sum(660, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b = 661

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(21, 61, 221, 661)

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 964

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 964)

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 964

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

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

ORIGINAL positive integer absolute magnitude=21>1

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

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

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

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

upf extract divisor3 occurrence0 = 7

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

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

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 21

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 := div(462, 21)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 22

Mobius inversion ORIGINAL inverse F(n/21):actual positive input=22;check complete integer range1..22

ALL positive divisors INCREASING=[1, 2, 11, 22];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(20, 1)

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

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor := mul(20, 22)

Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor = 440

Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b := sum(440, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b = 441

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum := sum(21, 41, 221, 441)

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 724

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) := mul(1, 724)

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 724

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:actual positive input for mu=22

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

ORIGINAL positive integer absolute magnitude=22>1

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

upf extract divisor2 occurrence0 := div(22, 2)

upf extract divisor2 occurrence0 = 11

next divisor=2;previous remaining=22;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 11)

upf canonical complete product factor1 = 22

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=22

complete increasing prime multiset=[2, 11];full product=22;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:input=22;COMPLETE prime multiset=[2, 11];distinct primes=[2, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 := div(462, 22)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 21

Mobius inversion ORIGINAL inverse F(n/22):actual positive input=21;check complete integer range1..21

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

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor := mul(20, 21)

Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor = 420

Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b := sum(420, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b = 421

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(21, 61, 141, 421)

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 644

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 644)

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 644

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:actual positive input for mu=33

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

ORIGINAL positive integer absolute magnitude=33>1

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

next divisor=2;previous remaining=33;all smaller divisors exhausted

extracted multiplicity=0;next remaining=33;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(33, 3)

upf extract divisor3 occurrence0 = 11

next divisor=3;previous remaining=33;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[3]

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

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 11)

upf canonical complete product factor1 = 33

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=33

complete increasing prime multiset=[3, 11];full product=33;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:input=33;COMPLETE prime multiset=[3, 11];distinct primes=[3, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 := div(462, 33)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 = 14

Mobius inversion ORIGINAL inverse F(n/33):actual positive input=14;check complete integer range1..14

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

Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor := mul(20, 14)

Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor = 280

Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b := sum(280, 1)

Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b = 281

Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum := sum(21, 41, 141, 281)

Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum = 484

Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) := mul(1, 484)

Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) = 484

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

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

ORIGINAL positive integer absolute magnitude=42>1

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

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

upf extract divisor2 occurrence0 = 21

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

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

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

upf extract divisor3 occurrence1 = 7

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

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

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

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

upf canonical complete product factor2 = 42

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 := div(462, 42)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 11

Mobius inversion ORIGINAL inverse F(n/42):actual positive input=11;check complete integer range1..11

ALL positive divisors INCREASING=[1, 11];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor := mul(20, 11)

Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor = 220

Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b := sum(220, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b = 221

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(21, 221)

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 242

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 242)

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -242

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:actual positive input for mu=66

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

ORIGINAL positive integer absolute magnitude=66>1

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

upf extract divisor2 occurrence0 := div(66, 2)

upf extract divisor2 occurrence0 = 33

next divisor=2;previous remaining=66;all smaller divisors exhausted

extracted multiplicity=1;next remaining=33;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(33, 3)

upf extract divisor3 occurrence1 = 11

next divisor=3;previous remaining=33;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo3=2;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, 11)

upf canonical complete product factor2 = 66

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=66

complete increasing prime multiset=[2, 3, 11];full product=66;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:input=66;COMPLETE prime multiset=[2, 3, 11];distinct primes=[2, 3, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 := div(462, 66)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 = 7

Mobius inversion ORIGINAL inverse F(n/66):actual positive input=7;check complete integer range1..7

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

Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor := mul(20, 7)

Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor = 140

Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b := sum(140, 1)

Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b = 141

Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum := sum(21, 141)

Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum = 162

Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) := mul(-1, 162)

Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) = -162

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor77:actual positive input for mu=77

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

ORIGINAL positive integer absolute magnitude=77>1

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

next divisor=2;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[]

next divisor=5;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[]

next divisor=6;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[]

upf extract divisor7 occurrence0 := div(77, 7)

upf extract divisor7 occurrence0 = 11

next divisor=7;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[7]

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

upf canonical complete product factor0 = 7

upf canonical complete product factor1 := mul(7, 11)

upf canonical complete product factor1 = 77

exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=77

complete increasing prime multiset=[7, 11];full product=77;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor77:input=77;COMPLETE prime multiset=[7, 11];distinct primes=[7, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 := div(462, 77)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 = 6

Mobius inversion ORIGINAL inverse F(n/77):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/77) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor := mul(20, 6)

Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum := sum(21, 41, 61, 121)

Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum = 244

Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) := mul(1, 244)

Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) = 244

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor154:actual positive input for mu=154

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

ORIGINAL positive integer absolute magnitude=154>1

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

upf extract divisor2 occurrence0 := div(154, 2)

upf extract divisor2 occurrence0 = 77

next divisor=2;previous remaining=154;all smaller divisors exhausted

extracted multiplicity=1;next remaining=77;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[2]

next divisor=5;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[2]

next divisor=6;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[2]

upf extract divisor7 occurrence1 := div(77, 7)

upf extract divisor7 occurrence1 = 11

next divisor=7;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[2, 7]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 7)

upf canonical complete product factor1 = 14

upf canonical complete product factor2 := mul(14, 11)

upf canonical complete product factor2 = 154

exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=154

complete increasing prime multiset=[2, 7, 11];full product=154;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor154:input=154;COMPLETE prime multiset=[2, 7, 11];distinct primes=[2, 7, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 := div(462, 154)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 = 3

Mobius inversion ORIGINAL inverse F(n/154):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/154) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor := mul(20, 3)

Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor = 60

Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b := sum(60, 1)

Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b = 61

Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum := sum(21, 61)

Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum = 82

Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) := mul(-1, 82)

Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) = -82

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor231:actual positive input for mu=231

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

ORIGINAL positive integer absolute magnitude=231>1

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

next divisor=2;previous remaining=231;all smaller divisors exhausted

extracted multiplicity=0;next remaining=231;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(231, 3)

upf extract divisor3 occurrence0 = 77

next divisor=3;previous remaining=231;all smaller divisors exhausted

extracted multiplicity=1;next remaining=77;remaining modulo3=2;extracted multiset=[3]

next divisor=4;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[3]

next divisor=5;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[3]

next divisor=6;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[3]

upf extract divisor7 occurrence1 := div(77, 7)

upf extract divisor7 occurrence1 = 11

next divisor=7;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[3, 7]

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 21

upf canonical complete product factor2 := mul(21, 11)

upf canonical complete product factor2 = 231

exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=231

complete increasing prime multiset=[3, 7, 11];full product=231;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor231:input=231;COMPLETE prime multiset=[3, 7, 11];distinct primes=[3, 7, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 := div(462, 231)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 = 2

Mobius inversion ORIGINAL inverse F(n/231):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/231) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor := mul(20, 2)

Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum := sum(21, 41)

Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum = 62

Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) := mul(-1, 62)

Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) = -62

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor462:actual positive input for mu=462

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

ORIGINAL positive integer absolute magnitude=462>1

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

upf extract divisor2 occurrence0 := div(462, 2)

upf extract divisor2 occurrence0 = 231

next divisor=2;previous remaining=462;all smaller divisors exhausted

extracted multiplicity=1;next remaining=231;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(231, 3)

upf extract divisor3 occurrence1 = 77

next divisor=3;previous remaining=231;all smaller divisors exhausted

extracted multiplicity=1;next remaining=77;remaining modulo3=2;extracted multiset=[2, 3]

next divisor=4;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[2, 3]

next divisor=5;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[2, 3]

next divisor=6;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[2, 3]

upf extract divisor7 occurrence2 := div(77, 7)

upf extract divisor7 occurrence2 = 11

next divisor=7;previous remaining=77;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[2, 3, 7]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

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

upf canonical complete product factor2 = 42

upf canonical complete product factor3 := mul(42, 11)

upf canonical complete product factor3 = 462

exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=462

complete increasing prime multiset=[2, 3, 7, 11];full product=462;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor462:input=462;COMPLETE prime multiset=[2, 3, 7, 11];distinct primes=[2, 3, 7, 11]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 := div(462, 462)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 = 1

Mobius inversion ORIGINAL inverse F(n/462):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/462) f(1) ORIGINAL a times divisor := mul(20, 1)

Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor = 20

Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b := sum(20, 1)

Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b = 21

Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum := sum(21)

Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum = 21

Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) := mul(1, 21)

Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) = 21

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(23056, -7688, -5768, 1924, -2888, -1928, 964, 724, 644, 484, -242, -162, 244, -82, -62, 21)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 9241

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(20, 462)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 9240

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(9240, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 9241

Mobius inversion ORIGINAL inverse:actual n=462;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 462, 23056, 23056], [2, -1, 231, 7688, -7688], [3, -1, 154, 5768, -5768], [6, 1, 77, 1924, 1924], [7, -1, 66, 2888, -2888], [11, -1, 42, 1928, -1928], [14, 1, 33, 964, 964], [21, 1, 22, 724, 724], [22, 1, 21, 644, 644], [33, 1, 14, 484, 484], [42, -1, 11, 242, -242], [66, -1, 7, 162, -162], [77, 1, 6, 244, 244], [154, -1, 3, 82, -82], [231, -1, 2, 62, -62], [462, 1, 1, 21, 21]]

complete reconstructed=9241;direct f(n)=9241;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(-7688, -5768, 1924, -2888, -1928, 964, 724, 644, 484, -242, -162, 244, -82, -62, 21)

Mobius inversion ORIGINAL omitted-divisor partial sum = -13815

Mobius inversion complete minus partial := sub(9241, -13815)

Mobius inversion complete minus partial = 23056

ORIGINAL full divisors=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462]

complete=9241=f(n)=9241;partial=-13815;lost term=23056!=0;incomplete_certificate fixed No

Reverse checks

• S385 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S384 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S383 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S382 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S381 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S380 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S379 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S378 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) with exact rational arithmetic, not rounded decimals.
• S377 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S376 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S375 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S374 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S373 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 with exact rational arithmetic, not rounded decimals.
• S372 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S371 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S370 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S369 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S368 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S367 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S366 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S365 · premise_and_result_audit — Recompute upf extract divisor7 occurrence2 with exact rational arithmetic, not rounded decimals.
• S364 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S363 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S362 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S361 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S360 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S359 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S358 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S357 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S356 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S355 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) with exact rational arithmetic, not rounded decimals.
• S354 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S353 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S352 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S351 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S350 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S349 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S348 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 with exact rational arithmetic, not rounded decimals.
• S347 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S346 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S345 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S344 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S343 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S342 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S341 · premise_and_result_audit — Recompute upf extract divisor7 occurrence1 with exact rational arithmetic, not rounded decimals.
• S340 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S339 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S338 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S337 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S336 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S335 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S334 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S333 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S332 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) with exact rational arithmetic, not rounded decimals.
• S331 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S330 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S329 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S328 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S327 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S326 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S325 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 with exact rational arithmetic, not rounded decimals.
• S324 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S323 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S322 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S321 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S320 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S319 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S318 · premise_and_result_audit — Recompute upf extract divisor7 occurrence1 with exact rational arithmetic, not rounded decimals.
• S317 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S316 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S315 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S314 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S313 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S312 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S311 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S310 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S309 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) with exact rational arithmetic, not rounded decimals.
• S308 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S307 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S306 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S305 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S304 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S303 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S302 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S301 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S300 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S299 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S298 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 with exact rational arithmetic, not rounded decimals.
• S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S296 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S295 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S294 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S293 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S292 · premise_and_result_audit — Recompute upf extract divisor7 occurrence0 with exact rational arithmetic, not rounded decimals.
• S291 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S290 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S289 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S288 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S287 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S286 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S285 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S284 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S280 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S278 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S277 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 with exact rational arithmetic, not rounded decimals.
• S276 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S275 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S274 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S272 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S270 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S268 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S266 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S265 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S260 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S255 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S253 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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(33) times SAME F(n/33) with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S235 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S229 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S225 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S214 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 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 canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S210 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S190 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S187 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S166 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S101 · 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.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · 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.
• S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · 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.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · 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.
• S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · 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.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(66) 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(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) 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(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(33) 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(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(22) 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(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) 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(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) 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(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(11) 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(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) 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

(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 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.