Question 22149: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=650, exact integer arithmetic function f(k)=a*k+b with a=24, 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=650, exact integer arithmetic function f(k)=a*k+b with a=24, 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=650;check complete integer range1..650

ALL positive divisors INCREASING=[1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650];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(650, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 650

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=650;check complete integer range1..650

ALL positive divisors INCREASING=[1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 48

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(24, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/1) f(25) ORIGINAL a times divisor := mul(24, 25)

Mobius inversion ORIGINAL inverse F(n/1) f(25) ORIGINAL a times divisor = 600

Mobius inversion ORIGINAL inverse F(n/1) f(25) plus ORIGINAL b := sum(600, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(25) plus ORIGINAL b = 601

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor := mul(24, 26)

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 624

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(624, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 625

Mobius inversion ORIGINAL inverse F(n/1) f(50) ORIGINAL a times divisor := mul(24, 50)

Mobius inversion ORIGINAL inverse F(n/1) f(50) ORIGINAL a times divisor = 1200

Mobius inversion ORIGINAL inverse F(n/1) f(50) plus ORIGINAL b := sum(1200, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(50) plus ORIGINAL b = 1201

Mobius inversion ORIGINAL inverse F(n/1) f(65) ORIGINAL a times divisor := mul(24, 65)

Mobius inversion ORIGINAL inverse F(n/1) f(65) ORIGINAL a times divisor = 1560

Mobius inversion ORIGINAL inverse F(n/1) f(65) plus ORIGINAL b := sum(1560, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(65) plus ORIGINAL b = 1561

Mobius inversion ORIGINAL inverse F(n/1) f(130) ORIGINAL a times divisor := mul(24, 130)

Mobius inversion ORIGINAL inverse F(n/1) f(130) ORIGINAL a times divisor = 3120

Mobius inversion ORIGINAL inverse F(n/1) f(130) plus ORIGINAL b := sum(3120, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(130) plus ORIGINAL b = 3121

Mobius inversion ORIGINAL inverse F(n/1) f(325) ORIGINAL a times divisor := mul(24, 325)

Mobius inversion ORIGINAL inverse F(n/1) f(325) ORIGINAL a times divisor = 7800

Mobius inversion ORIGINAL inverse F(n/1) f(325) plus ORIGINAL b := sum(7800, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(325) plus ORIGINAL b = 7801

Mobius inversion ORIGINAL inverse F(n/1) f(650) ORIGINAL a times divisor := mul(24, 650)

Mobius inversion ORIGINAL inverse F(n/1) f(650) ORIGINAL a times divisor = 15600

Mobius inversion ORIGINAL inverse F(n/1) f(650) plus ORIGINAL b := sum(15600, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(650) plus ORIGINAL b = 15601

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(25, 49, 121, 241, 313, 601, 625, 1201, 1561, 3121, 7801, 15601)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 31260

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 31260)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 31260

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(650, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 325

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=325;check complete integer range1..325

ALL positive divisors INCREASING=[1, 5, 13, 25, 65, 325];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/2) f(25) ORIGINAL a times divisor := mul(24, 25)

Mobius inversion ORIGINAL inverse F(n/2) f(25) ORIGINAL a times divisor = 600

Mobius inversion ORIGINAL inverse F(n/2) f(25) plus ORIGINAL b := sum(600, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(25) plus ORIGINAL b = 601

Mobius inversion ORIGINAL inverse F(n/2) f(65) ORIGINAL a times divisor := mul(24, 65)

Mobius inversion ORIGINAL inverse F(n/2) f(65) ORIGINAL a times divisor = 1560

Mobius inversion ORIGINAL inverse F(n/2) f(65) plus ORIGINAL b := sum(1560, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(65) plus ORIGINAL b = 1561

Mobius inversion ORIGINAL inverse F(n/2) f(325) ORIGINAL a times divisor := mul(24, 325)

Mobius inversion ORIGINAL inverse F(n/2) f(325) ORIGINAL a times divisor = 7800

Mobius inversion ORIGINAL inverse F(n/2) f(325) plus ORIGINAL b := sum(7800, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(325) plus ORIGINAL b = 7801

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(25, 121, 313, 601, 1561, 7801)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 10422

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 10422)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -10422

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(650, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 130

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=130;check complete integer range1..130

ALL positive divisors INCREASING=[1, 2, 5, 10, 13, 26, 65, 130];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/5) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/5) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/5) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/5) f(10) ORIGINAL a times divisor := mul(24, 10)

Mobius inversion ORIGINAL inverse F(n/5) f(10) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/5) f(10) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(10) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/5) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/5) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/5) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/5) f(26) ORIGINAL a times divisor := mul(24, 26)

Mobius inversion ORIGINAL inverse F(n/5) f(26) ORIGINAL a times divisor = 624

Mobius inversion ORIGINAL inverse F(n/5) f(26) plus ORIGINAL b := sum(624, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(26) plus ORIGINAL b = 625

Mobius inversion ORIGINAL inverse F(n/5) f(65) ORIGINAL a times divisor := mul(24, 65)

Mobius inversion ORIGINAL inverse F(n/5) f(65) ORIGINAL a times divisor = 1560

Mobius inversion ORIGINAL inverse F(n/5) f(65) plus ORIGINAL b := sum(1560, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(65) plus ORIGINAL b = 1561

Mobius inversion ORIGINAL inverse F(n/5) f(130) ORIGINAL a times divisor := mul(24, 130)

Mobius inversion ORIGINAL inverse F(n/5) f(130) ORIGINAL a times divisor = 3120

Mobius inversion ORIGINAL inverse F(n/5) f(130) plus ORIGINAL b := sum(3120, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(130) plus ORIGINAL b = 3121

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(25, 49, 121, 241, 313, 625, 1561, 3121)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 6056

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 6056)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -6056

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(650, 10)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 65

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=65;check complete integer range1..65

ALL positive divisors INCREASING=[1, 5, 13, 65];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/10) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/10) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/10) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/10) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/10) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/10) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/10) f(65) ORIGINAL a times divisor := mul(24, 65)

Mobius inversion ORIGINAL inverse F(n/10) f(65) ORIGINAL a times divisor = 1560

Mobius inversion ORIGINAL inverse F(n/10) f(65) plus ORIGINAL b := sum(1560, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(65) plus ORIGINAL b = 1561

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(25, 121, 313, 1561)

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 2020

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 2020)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 2020

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

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

ORIGINAL positive integer absolute magnitude=13>1

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

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

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

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

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

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

upf canonical complete product factor0 = 13

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 := div(650, 13)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 50

Mobius inversion ORIGINAL inverse F(n/13):actual positive input=50;check complete integer range1..50

ALL positive divisors INCREASING=[1, 2, 5, 10, 25, 50];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/13) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/13) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/13) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/13) f(10) ORIGINAL a times divisor := mul(24, 10)

Mobius inversion ORIGINAL inverse F(n/13) f(10) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/13) f(10) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(10) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/13) f(25) ORIGINAL a times divisor := mul(24, 25)

Mobius inversion ORIGINAL inverse F(n/13) f(25) ORIGINAL a times divisor = 600

Mobius inversion ORIGINAL inverse F(n/13) f(25) plus ORIGINAL b := sum(600, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(25) plus ORIGINAL b = 601

Mobius inversion ORIGINAL inverse F(n/13) f(50) ORIGINAL a times divisor := mul(24, 50)

Mobius inversion ORIGINAL inverse F(n/13) f(50) ORIGINAL a times divisor = 1200

Mobius inversion ORIGINAL inverse F(n/13) f(50) plus ORIGINAL b := sum(1200, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(50) plus ORIGINAL b = 1201

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(25, 49, 121, 241, 601, 1201)

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 2238

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 2238)

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -2238

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor25:actual positive input for mu=25

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

ORIGINAL positive integer absolute magnitude=25>1

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

next divisor=2;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo4=1;extracted multiset=[]

upf extract divisor5 occurrence0 := div(25, 5)

upf extract divisor5 occurrence0 = 5

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

upf extract divisor5 occurrence1 = 1

next divisor=5;previous remaining=25;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 5

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

upf canonical complete product factor1 = 25

exhausted trial divisors below6;terminal remaining=1;d*d>remaining;ORIGINAL n=25

complete increasing prime multiset=[5, 5];full product=25;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor25:input=25;COMPLETE prime multiset=[5, 5];distinct primes=[5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor25 := div(650, 25)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor25 = 26

Mobius inversion ORIGINAL inverse F(n/25):actual positive input=26;check complete integer range1..26

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

Mobius inversion ORIGINAL inverse F(n/25) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/25) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/25) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/25) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/25) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/25) f(2) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/25) f(2) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/25) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/25) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/25) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/25) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/25) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/25) f(26) ORIGINAL a times divisor := mul(24, 26)

Mobius inversion ORIGINAL inverse F(n/25) f(26) ORIGINAL a times divisor = 624

Mobius inversion ORIGINAL inverse F(n/25) f(26) plus ORIGINAL b := sum(624, 1)

Mobius inversion ORIGINAL inverse F(n/25) f(26) plus ORIGINAL b = 625

Mobius inversion ORIGINAL inverse F(n/25) full forward divisor sum := sum(25, 49, 313, 625)

Mobius inversion ORIGINAL inverse F(n/25) full forward divisor sum = 1012

Mobius inversion ORIGINAL inverse mu(25) times SAME F(n/25) := mul(0, 1012)

Mobius inversion ORIGINAL inverse mu(25) times SAME F(n/25) = 0

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

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

ORIGINAL positive integer absolute magnitude=26>1

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

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

upf extract divisor2 occurrence0 = 13

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

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

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

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

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 13)

upf canonical complete product factor1 = 26

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 := div(650, 26)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 25

Mobius inversion ORIGINAL inverse F(n/26):actual positive input=25;check complete integer range1..25

ALL positive divisors INCREASING=[1, 5, 25];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/26) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/26) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/26) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/26) f(25) ORIGINAL a times divisor := mul(24, 25)

Mobius inversion ORIGINAL inverse F(n/26) f(25) ORIGINAL a times divisor = 600

Mobius inversion ORIGINAL inverse F(n/26) f(25) plus ORIGINAL b := sum(600, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(25) plus ORIGINAL b = 601

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum := sum(25, 121, 601)

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 747

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) := mul(1, 747)

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 747

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor50:actual positive input for mu=50

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

ORIGINAL positive integer absolute magnitude=50>1

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

upf extract divisor2 occurrence0 := div(50, 2)

upf extract divisor2 occurrence0 = 25

next divisor=2;previous remaining=50;all smaller divisors exhausted

extracted multiplicity=1;next remaining=25;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo4=1;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(25, 5)

upf extract divisor5 occurrence1 = 5

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

upf extract divisor5 occurrence2 = 1

next divisor=5;previous remaining=25;all smaller divisors exhausted

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

upf canonical complete product factor2 = 50

exhausted trial divisors below6;terminal remaining=1;d*d>remaining;ORIGINAL n=50

complete increasing prime multiset=[2, 5, 5];full product=50;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor50:input=50;COMPLETE prime multiset=[2, 5, 5];distinct primes=[2, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor50 := div(650, 50)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor50 = 13

Mobius inversion ORIGINAL inverse F(n/50):actual positive input=13;check complete integer range1..13

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

Mobius inversion ORIGINAL inverse F(n/50) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/50) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/50) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/50) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/50) f(13) ORIGINAL a times divisor := mul(24, 13)

Mobius inversion ORIGINAL inverse F(n/50) f(13) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/50) f(13) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/50) f(13) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/50) full forward divisor sum := sum(25, 313)

Mobius inversion ORIGINAL inverse F(n/50) full forward divisor sum = 338

Mobius inversion ORIGINAL inverse mu(50) times SAME F(n/50) := mul(0, 338)

Mobius inversion ORIGINAL inverse mu(50) times SAME F(n/50) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor65:actual positive input for mu=65

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

ORIGINAL positive integer absolute magnitude=65>1

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

next divisor=2;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=0;next remaining=65;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=0;next remaining=65;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=0;next remaining=65;remaining modulo4=1;extracted multiset=[]

upf extract divisor5 occurrence0 := div(65, 5)

upf extract divisor5 occurrence0 = 13

next divisor=5;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo5=3;extracted multiset=[5]

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

upf canonical complete product factor0 = 5

upf canonical complete product factor1 := mul(5, 13)

upf canonical complete product factor1 = 65

exhausted trial divisors below6;terminal remaining=13;d*d>remaining;ORIGINAL n=65

complete increasing prime multiset=[5, 13];full product=65;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor65:input=65;COMPLETE prime multiset=[5, 13];distinct primes=[5, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor65 := div(650, 65)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor65 = 10

Mobius inversion ORIGINAL inverse F(n/65):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/65) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/65) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/65) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/65) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/65) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/65) f(2) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/65) f(2) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/65) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/65) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/65) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/65) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/65) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/65) f(10) ORIGINAL a times divisor := mul(24, 10)

Mobius inversion ORIGINAL inverse F(n/65) f(10) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/65) f(10) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/65) f(10) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/65) full forward divisor sum := sum(25, 49, 121, 241)

Mobius inversion ORIGINAL inverse F(n/65) full forward divisor sum = 436

Mobius inversion ORIGINAL inverse mu(65) times SAME F(n/65) := mul(1, 436)

Mobius inversion ORIGINAL inverse mu(65) times SAME F(n/65) = 436

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor130:actual positive input for mu=130

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

ORIGINAL positive integer absolute magnitude=130>1

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

upf extract divisor2 occurrence0 := div(130, 2)

upf extract divisor2 occurrence0 = 65

next divisor=2;previous remaining=130;all smaller divisors exhausted

extracted multiplicity=1;next remaining=65;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=0;next remaining=65;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=0;next remaining=65;remaining modulo4=1;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(65, 5)

upf extract divisor5 occurrence1 = 13

next divisor=5;previous remaining=65;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo5=3;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, 13)

upf canonical complete product factor2 = 130

exhausted trial divisors below6;terminal remaining=13;d*d>remaining;ORIGINAL n=130

complete increasing prime multiset=[2, 5, 13];full product=130;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor130:input=130;COMPLETE prime multiset=[2, 5, 13];distinct primes=[2, 5, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor130 := div(650, 130)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor130 = 5

Mobius inversion ORIGINAL inverse F(n/130):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/130) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/130) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/130) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/130) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/130) f(5) ORIGINAL a times divisor := mul(24, 5)

Mobius inversion ORIGINAL inverse F(n/130) f(5) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/130) f(5) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/130) f(5) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/130) full forward divisor sum := sum(25, 121)

Mobius inversion ORIGINAL inverse F(n/130) full forward divisor sum = 146

Mobius inversion ORIGINAL inverse mu(130) times SAME F(n/130) := mul(-1, 146)

Mobius inversion ORIGINAL inverse mu(130) times SAME F(n/130) = -146

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor325:actual positive input for mu=325

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

ORIGINAL positive integer absolute magnitude=325>1

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

next divisor=2;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=0;next remaining=325;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=0;next remaining=325;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=0;next remaining=325;remaining modulo4=1;extracted multiset=[]

upf extract divisor5 occurrence0 := div(325, 5)

upf extract divisor5 occurrence0 = 65

upf extract divisor5 occurrence1 := div(65, 5)

upf extract divisor5 occurrence1 = 13

next divisor=5;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=2;next remaining=13;remaining modulo5=3;extracted multiset=[5, 5]

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

upf canonical complete product factor0 = 5

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

upf canonical complete product factor1 = 25

upf canonical complete product factor2 := mul(25, 13)

upf canonical complete product factor2 = 325

exhausted trial divisors below6;terminal remaining=13;d*d>remaining;ORIGINAL n=325

complete increasing prime multiset=[5, 5, 13];full product=325;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor325:input=325;COMPLETE prime multiset=[5, 5, 13];distinct primes=[5, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor325 := div(650, 325)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor325 = 2

Mobius inversion ORIGINAL inverse F(n/325):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/325) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/325) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/325) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/325) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/325) f(2) ORIGINAL a times divisor := mul(24, 2)

Mobius inversion ORIGINAL inverse F(n/325) f(2) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/325) f(2) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/325) f(2) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/325) full forward divisor sum := sum(25, 49)

Mobius inversion ORIGINAL inverse F(n/325) full forward divisor sum = 74

Mobius inversion ORIGINAL inverse mu(325) times SAME F(n/325) := mul(0, 74)

Mobius inversion ORIGINAL inverse mu(325) times SAME F(n/325) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor650:actual positive input for mu=650

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

ORIGINAL positive integer absolute magnitude=650>1

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

upf extract divisor2 occurrence0 := div(650, 2)

upf extract divisor2 occurrence0 = 325

next divisor=2;previous remaining=650;all smaller divisors exhausted

extracted multiplicity=1;next remaining=325;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=0;next remaining=325;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=0;next remaining=325;remaining modulo4=1;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(325, 5)

upf extract divisor5 occurrence1 = 65

upf extract divisor5 occurrence2 := div(65, 5)

upf extract divisor5 occurrence2 = 13

next divisor=5;previous remaining=325;all smaller divisors exhausted

extracted multiplicity=2;next remaining=13;remaining modulo5=3;extracted multiset=[2, 5, 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, 5)

upf canonical complete product factor2 = 50

upf canonical complete product factor3 := mul(50, 13)

upf canonical complete product factor3 = 650

exhausted trial divisors below6;terminal remaining=13;d*d>remaining;ORIGINAL n=650

complete increasing prime multiset=[2, 5, 5, 13];full product=650;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor650:input=650;COMPLETE prime multiset=[2, 5, 5, 13];distinct primes=[2, 5, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor650 := div(650, 650)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor650 = 1

Mobius inversion ORIGINAL inverse F(n/650):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/650) f(1) ORIGINAL a times divisor := mul(24, 1)

Mobius inversion ORIGINAL inverse F(n/650) f(1) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/650) f(1) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/650) f(1) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/650) full forward divisor sum := sum(25)

Mobius inversion ORIGINAL inverse F(n/650) full forward divisor sum = 25

Mobius inversion ORIGINAL inverse mu(650) times SAME F(n/650) := mul(0, 25)

Mobius inversion ORIGINAL inverse mu(650) times SAME F(n/650) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(31260, -10422, -6056, 2020, -2238, 0, 747, 0, 436, -146, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 15601

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(24, 650)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 15600

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(15600, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 15601

Mobius inversion ORIGINAL inverse:actual n=650;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 650, 31260, 31260], [2, -1, 325, 10422, -10422], [5, -1, 130, 6056, -6056], [10, 1, 65, 2020, 2020], [13, -1, 50, 2238, -2238], [25, 0, 26, 1012, 0], [26, 1, 25, 747, 747], [50, 0, 13, 338, 0], [65, 1, 10, 436, 436], [130, -1, 5, 146, -146], [325, 0, 2, 74, 0], [650, 0, 1, 25, 0]]

complete reconstructed=15601;direct f(n)=15601;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(-10422, -6056, 2020, -2238, 0, 747, 0, 436, -146, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -15659

Mobius inversion complete minus partial := sub(15601, -15659)

Mobius inversion complete minus partial = 31260

ORIGINAL full divisors=[1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650]

complete=15601=f(n)=15601;partial=-15659;lost term=31260!=0;incomplete_certificate fixed No

Reverse checks

• S280 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S279 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S278 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S277 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S276 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S275 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S274 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(650) times SAME F(n/650) with exact rational arithmetic, not rounded decimals.
• S272 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/650) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/650) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/650) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor650 with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S266 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S265 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S260 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S257 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S255 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S253 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S252 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(325) times SAME F(n/325) with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/325) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S250 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/325) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/325) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/325) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/325) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor325 with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S242 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S238 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S235 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S231 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(130) times SAME F(n/130) with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/130) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/130) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S228 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/130) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/130) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/130) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor130 with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S221 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S217 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S213 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S210 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(65) times SAME F(n/65) with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S206 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/65) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor65 with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S196 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S193 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S190 · premise_and_result_audit — 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S187 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(50) times SAME F(n/50) with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/50) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/50) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/50) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/50) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/50) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor50 with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S177 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S173 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — 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 extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(25) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S153 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(25) times SAME F(n/25) with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/25) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor25 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(50) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(50) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(25) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(65) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(65) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S090 · 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.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(130) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(130) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(65) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(65) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S064 · 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.
• S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S062 · 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.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(325) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(325) 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(65) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(65) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(25) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S042 · 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.
• S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(650) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(650) 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(325) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(325) 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(130) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(130) 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(65) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(65) 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(50) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(50) 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(26) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(25) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(25) 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(13) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S013 · 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.
• S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S011 · 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.
• 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

(650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, 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.