Question 22138: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=210, exact integer arithmetic function f(k)=a*k+b with a=13, 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=210, exact integer arithmetic function f(k)=a*k+b with a=13, 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=210;check complete integer range1..210

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210];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(210, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 210

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=210;check complete integer range1..210

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210];not only primes/proper divisors or all integers<=n

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 26

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(39, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(13, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 78

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(78, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 79

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(13, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 130

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(130, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 131

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor := mul(13, 14)

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 182

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b := sum(182, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 183

Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor := mul(13, 15)

Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor = 195

Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b := sum(195, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b = 196

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor := mul(13, 21)

Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 273

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b := sum(273, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 274

Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor := mul(13, 30)

Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor = 390

Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b := sum(390, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b = 391

Mobius inversion ORIGINAL inverse F(n/1) f(35) ORIGINAL a times divisor := mul(13, 35)

Mobius inversion ORIGINAL inverse F(n/1) f(35) ORIGINAL a times divisor = 455

Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b := sum(455, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b = 456

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor := mul(13, 42)

Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 546

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b := sum(546, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 547

Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor := mul(13, 70)

Mobius inversion ORIGINAL inverse F(n/1) f(70) ORIGINAL a times divisor = 910

Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b := sum(910, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(70) plus ORIGINAL b = 911

Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor := mul(13, 105)

Mobius inversion ORIGINAL inverse F(n/1) f(105) ORIGINAL a times divisor = 1365

Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b := sum(1365, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(105) plus ORIGINAL b = 1366

Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor := mul(13, 210)

Mobius inversion ORIGINAL inverse F(n/1) f(210) ORIGINAL a times divisor = 2730

Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b := sum(2730, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(210) plus ORIGINAL b = 2731

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(14, 27, 40, 66, 79, 92, 131, 183, 196, 274, 391, 456, 547, 911, 1366, 2731)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 7504

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 7504)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 7504

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(210, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 105

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=105;check complete integer range1..105

ALL positive divisors INCREASING=[1, 3, 5, 7, 15, 21, 35, 105];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 13

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

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(39, 1)

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

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor := mul(13, 15)

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor = 195

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b := sum(195, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b = 196

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor := mul(13, 21)

Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 273

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b := sum(273, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 274

Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor := mul(13, 35)

Mobius inversion ORIGINAL inverse F(n/2) f(35) ORIGINAL a times divisor = 455

Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b := sum(455, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(35) plus ORIGINAL b = 456

Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor := mul(13, 105)

Mobius inversion ORIGINAL inverse F(n/2) f(105) ORIGINAL a times divisor = 1365

Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b := sum(1365, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(105) plus ORIGINAL b = 1366

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(14, 40, 66, 92, 196, 274, 456, 1366)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 2504

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 2504)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -2504

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(210, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 70

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=70;check complete integer range1..70

ALL positive divisors INCREASING=[1, 2, 5, 7, 10, 14, 35, 70];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 13

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

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

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor := mul(13, 10)

Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor = 130

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b := sum(130, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b = 131

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(13, 14)

Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 182

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b := sum(182, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 183

Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor := mul(13, 35)

Mobius inversion ORIGINAL inverse F(n/3) f(35) ORIGINAL a times divisor = 455

Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b := sum(455, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(35) plus ORIGINAL b = 456

Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor := mul(13, 70)

Mobius inversion ORIGINAL inverse F(n/3) f(70) ORIGINAL a times divisor = 910

Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b := sum(910, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(70) plus ORIGINAL b = 911

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(14, 27, 66, 92, 131, 183, 456, 911)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 1880

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 1880)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -1880

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:actual positive input for mu=5

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

ORIGINAL positive integer absolute magnitude=5>1

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

next divisor=2;previous remaining=5;all smaller divisors exhausted

extracted multiplicity=0;next remaining=5;remaining modulo2=1;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 5)

upf canonical complete product factor0 = 5

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=5

complete increasing prime multiset=[5];full product=5;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:input=5;COMPLETE prime multiset=[5];distinct primes=[5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(210, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 42

Mobius inversion ORIGINAL inverse F(n/5):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/5) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor := mul(13, 6)

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor = 78

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b := sum(78, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b = 79

Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor := mul(13, 14)

Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor = 182

Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b := sum(182, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b = 183

Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor := mul(13, 21)

Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor = 273

Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b := sum(273, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b = 274

Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor := mul(13, 42)

Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor = 546

Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b := sum(546, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b = 547

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(14, 27, 40, 79, 92, 183, 274, 547)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 1256

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 1256)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -1256

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(210, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 35

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=35;check complete integer range1..35

ALL positive divisors INCREASING=[1, 5, 7, 35];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 13

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

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor := mul(13, 35)

Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor = 455

Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b := sum(455, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b = 456

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(14, 66, 92, 456)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 628

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 628)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 628

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(210, 7)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 30

Mobius inversion ORIGINAL inverse F(n/7):actual positive input=30;check complete integer range1..30

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 10, 15, 30];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor := mul(13, 6)

Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 78

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b := sum(78, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 79

Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor := mul(13, 10)

Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor = 130

Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b := sum(130, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b = 131

Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor := mul(13, 15)

Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor = 195

Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b := sum(195, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b = 196

Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor := mul(13, 30)

Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor = 390

Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b := sum(390, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b = 391

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(14, 27, 40, 66, 79, 131, 196, 391)

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 944

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 944)

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -944

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:actual positive input for mu=10

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

ORIGINAL positive integer absolute magnitude=10>1

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

upf extract divisor2 occurrence0 := div(10, 2)

upf extract divisor2 occurrence0 = 5

next divisor=2;previous remaining=10;all smaller divisors exhausted

extracted multiplicity=1;next remaining=5;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, 5)

upf canonical complete product factor1 = 10

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=10

complete increasing prime multiset=[2, 5];full product=10;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:input=10;COMPLETE prime multiset=[2, 5];distinct primes=[2, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 := div(210, 10)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 21

Mobius inversion ORIGINAL inverse F(n/10):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/10) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor := mul(13, 21)

Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor = 273

Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b := sum(273, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b = 274

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(14, 40, 92, 274)

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 420

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 420)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 420

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(210, 14)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 15

Mobius inversion ORIGINAL inverse F(n/14):actual positive input=15;check complete integer range1..15

ALL positive divisors INCREASING=[1, 3, 5, 15];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(13, 1)

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

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor := mul(13, 15)

Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor = 195

Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b := sum(195, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b = 196

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(14, 40, 66, 196)

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 316

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 316)

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 316

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:actual positive input for mu=15

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

ORIGINAL positive integer absolute magnitude=15>1

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

next divisor=2;previous remaining=15;all smaller divisors exhausted

extracted multiplicity=0;next remaining=15;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(15, 3)

upf extract divisor3 occurrence0 = 5

next divisor=3;previous remaining=15;all smaller divisors exhausted

extracted multiplicity=1;next remaining=5;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, 5)

upf canonical complete product factor1 = 15

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=15

complete increasing prime multiset=[3, 5];full product=15;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor15:input=15;COMPLETE prime multiset=[3, 5];distinct primes=[3, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 := div(210, 15)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 = 14

Mobius inversion ORIGINAL inverse F(n/15):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/15) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor := mul(13, 14)

Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor = 182

Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b := sum(182, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b = 183

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum := sum(14, 27, 92, 183)

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum = 316

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) := mul(1, 316)

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) = 316

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(210, 21)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 10

Mobius inversion ORIGINAL inverse F(n/21):actual positive input=10;check complete integer range1..10

ALL positive divisors INCREASING=[1, 2, 5, 10];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(13, 1)

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

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor := mul(13, 10)

Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor = 130

Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b := sum(130, 1)

Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b = 131

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum := sum(14, 27, 66, 131)

Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 238

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) := mul(1, 238)

Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 238

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:actual positive input for mu=30

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

ORIGINAL positive integer absolute magnitude=30>1

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

upf extract divisor2 occurrence0 := div(30, 2)

upf extract divisor2 occurrence0 = 15

next divisor=2;previous remaining=30;all smaller divisors exhausted

extracted multiplicity=1;next remaining=15;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(15, 3)

upf extract divisor3 occurrence1 = 5

next divisor=3;previous remaining=15;all smaller divisors exhausted

extracted multiplicity=1;next remaining=5;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, 5)

upf canonical complete product factor2 = 30

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=30

complete increasing prime multiset=[2, 3, 5];full product=30;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor30:input=30;COMPLETE prime multiset=[2, 3, 5];distinct primes=[2, 3, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 := div(210, 30)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 = 7

Mobius inversion ORIGINAL inverse F(n/30):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/30) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor := mul(13, 7)

Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor = 91

Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b := sum(91, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b = 92

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum := sum(14, 92)

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum = 106

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) := mul(-1, 106)

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) = -106

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:actual positive input for mu=35

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

ORIGINAL positive integer absolute magnitude=35>1

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

next divisor=2;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[]

upf extract divisor5 occurrence0 := div(35, 5)

upf extract divisor5 occurrence0 = 7

next divisor=5;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[5]

upf canonical complete product factor0 := mul(1, 5)

upf canonical complete product factor0 = 5

upf canonical complete product factor1 := mul(5, 7)

upf canonical complete product factor1 = 35

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=35

complete increasing prime multiset=[5, 7];full product=35;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor35:input=35;COMPLETE prime multiset=[5, 7];distinct primes=[5, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 := div(210, 35)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 = 6

Mobius inversion ORIGINAL inverse F(n/35):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/35) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor := mul(13, 6)

Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor = 78

Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b := sum(78, 1)

Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b = 79

Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum := sum(14, 27, 40, 79)

Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum = 160

Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) := mul(1, 160)

Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) = 160

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(210, 42)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 5

Mobius inversion ORIGINAL inverse F(n/42):actual positive input=5;check complete integer range1..5

ALL positive divisors INCREASING=[1, 5];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor := mul(13, 5)

Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor = 65

Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b := sum(65, 1)

Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b = 66

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(14, 66)

Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 80

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 80)

Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -80

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:actual positive input for mu=70

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

ORIGINAL positive integer absolute magnitude=70>1

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

upf extract divisor2 occurrence0 := div(70, 2)

upf extract divisor2 occurrence0 = 35

next divisor=2;previous remaining=70;all smaller divisors exhausted

extracted multiplicity=1;next remaining=35;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(35, 5)

upf extract divisor5 occurrence1 = 7

next divisor=5;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 5]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 5)

upf canonical complete product factor1 = 10

upf canonical complete product factor2 := mul(10, 7)

upf canonical complete product factor2 = 70

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=70

complete increasing prime multiset=[2, 5, 7];full product=70;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor70:input=70;COMPLETE prime multiset=[2, 5, 7];distinct primes=[2, 5, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 := div(210, 70)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 = 3

Mobius inversion ORIGINAL inverse F(n/70):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/70) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor := mul(13, 3)

Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b := sum(39, 1)

Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b = 40

Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum := sum(14, 40)

Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum = 54

Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) := mul(-1, 54)

Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) = -54

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:actual positive input for mu=105

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

ORIGINAL positive integer absolute magnitude=105>1

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

next divisor=2;previous remaining=105;all smaller divisors exhausted

extracted multiplicity=0;next remaining=105;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(105, 3)

upf extract divisor3 occurrence0 = 35

next divisor=3;previous remaining=105;all smaller divisors exhausted

extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[3]

next divisor=4;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[3]

upf extract divisor5 occurrence1 := div(35, 5)

upf extract divisor5 occurrence1 = 7

next divisor=5;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[3, 5]

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

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 5)

upf canonical complete product factor1 = 15

upf canonical complete product factor2 := mul(15, 7)

upf canonical complete product factor2 = 105

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=105

complete increasing prime multiset=[3, 5, 7];full product=105;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor105:input=105;COMPLETE prime multiset=[3, 5, 7];distinct primes=[3, 5, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 := div(210, 105)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 = 2

Mobius inversion ORIGINAL inverse F(n/105):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/105) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor := mul(13, 2)

Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor = 26

Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b := sum(26, 1)

Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b = 27

Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum := sum(14, 27)

Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum = 41

Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) := mul(-1, 41)

Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) = -41

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:actual positive input for mu=210

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

ORIGINAL positive integer absolute magnitude=210>1

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

upf extract divisor2 occurrence0 := div(210, 2)

upf extract divisor2 occurrence0 = 105

next divisor=2;previous remaining=210;all smaller divisors exhausted

extracted multiplicity=1;next remaining=105;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(105, 3)

upf extract divisor3 occurrence1 = 35

next divisor=3;previous remaining=105;all smaller divisors exhausted

extracted multiplicity=1;next remaining=35;remaining modulo3=2;extracted multiset=[2, 3]

next divisor=4;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=0;next remaining=35;remaining modulo4=3;extracted multiset=[2, 3]

upf extract divisor5 occurrence2 := div(35, 5)

upf extract divisor5 occurrence2 = 7

next divisor=5;previous remaining=35;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;remaining modulo5=2;extracted multiset=[2, 3, 5]

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, 5)

upf canonical complete product factor2 = 30

upf canonical complete product factor3 := mul(30, 7)

upf canonical complete product factor3 = 210

exhausted trial divisors below6;terminal remaining=7;d*d>remaining;ORIGINAL n=210

complete increasing prime multiset=[2, 3, 5, 7];full product=210;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor210:input=210;COMPLETE prime multiset=[2, 3, 5, 7];distinct primes=[2, 3, 5, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 := div(210, 210)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 = 1

Mobius inversion ORIGINAL inverse F(n/210):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/210) f(1) ORIGINAL a times divisor := mul(13, 1)

Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor = 13

Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b := sum(13, 1)

Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b = 14

Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum := sum(14)

Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum = 14

Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) := mul(1, 14)

Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) = 14

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(7504, -2504, -1880, -1256, 628, -944, 420, 316, 316, 238, -106, 160, -80, -54, -41, 14)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 2731

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(13, 210)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 2730

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(2730, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 2731

Mobius inversion ORIGINAL inverse:actual n=210;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 210, 7504, 7504], [2, -1, 105, 2504, -2504], [3, -1, 70, 1880, -1880], [5, -1, 42, 1256, -1256], [6, 1, 35, 628, 628], [7, -1, 30, 944, -944], [10, 1, 21, 420, 420], [14, 1, 15, 316, 316], [15, 1, 14, 316, 316], [21, 1, 10, 238, 238], [30, -1, 7, 106, -106], [35, 1, 6, 160, 160], [42, -1, 5, 80, -80], [70, -1, 3, 54, -54], [105, -1, 2, 41, -41], [210, 1, 1, 14, 14]]

complete reconstructed=2731;direct f(n)=2731;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(-2504, -1880, -1256, 628, -944, 420, 316, 316, 238, -106, 160, -80, -54, -41, 14)

Mobius inversion ORIGINAL omitted-divisor partial sum = -4773

Mobius inversion complete minus partial := sub(2731, -4773)

Mobius inversion complete minus partial = 7504

ORIGINAL full divisors=[1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]

complete=2731=f(n)=2731;partial=-4773;lost term=7504!=0;incomplete_certificate fixed No

Reverse checks

• S375 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S374 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S373 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum 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 — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S370 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S369 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S368 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(210) times SAME F(n/210) with exact rational arithmetic, not rounded decimals.
• S367 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S366 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S365 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/210) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor210 with exact rational arithmetic, not rounded decimals.
• 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 canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S359 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S358 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S357 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
• S354 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S353 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S352 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S351 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S350 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S347 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(105) times SAME F(n/105) with exact rational arithmetic, not rounded decimals.
• S346 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S345 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S344 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S343 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S342 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/105) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S341 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S340 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor105 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S336 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S335 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S332 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S331 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S330 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S329 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S328 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S327 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S326 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(70) times SAME F(n/70) with exact rational arithmetic, not rounded decimals.
• S325 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S324 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S323 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S322 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S321 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/70) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S320 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S319 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor70 with exact rational arithmetic, not rounded decimals.
• S318 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S315 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S314 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 divisor5 occurrence1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S308 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S306 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S305 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) with exact rational arithmetic, not rounded decimals.
• S304 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S303 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S302 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S301 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S300 · 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.
• 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 divisor42 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 factor2 with exact rational arithmetic, not rounded decimals.
• S294 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S293 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S292 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S291 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Mobius inversion ORIGINAL inverse mu(35) times SAME F(n/35) with exact rational arithmetic, not rounded decimals.
• S285 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S284 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S280 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S278 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S277 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/35) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor35 with exact rational arithmetic, not rounded decimals.
• S274 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S272 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S269 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S266 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S265 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S264 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S260 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S254 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S253 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 divisor3 occurrence1 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 — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S245 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S237 · 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.
• S236 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S235 · 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.
• 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 with exact rational arithmetic, not rounded decimals.
• S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S230 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S214 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S212 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S209 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S195 · 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.
• S194 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S193 · 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.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S188 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(30) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S153 · 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.
• S152 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S149 · 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.
• S148 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S147 · 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.
• S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S145 · 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.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(35) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S131 · 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.
• S130 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S127 · 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.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — 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(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(70) 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(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(35) 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(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S083 · 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.
• S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) 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(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S079 · 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.
• S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) 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(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(105) 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(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(35) 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(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S058 · 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.
• S057 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(15) 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(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S054 · 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.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) 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(210) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(210) 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(105) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(105) 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(70) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(70) 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(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S031 · 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.
• S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(35) 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(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(30) 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(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · 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.
• S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) 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(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · 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.
• S020 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) 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(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · 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.
• S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) 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

(210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, 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.