Question 22133: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=90, exact integer arithmetic function f(k)=a*k+b with a=8, 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=90, exact integer arithmetic function f(k)=a*k+b with a=8, 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=90;check complete integer range1..90

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90];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(90, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 90

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=90;check complete integer range1..90

ALL positive divisors INCREASING=[1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(8, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(8, 2)

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(8, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 48

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

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor := mul(8, 9)

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(8, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 80

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(80, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 81

Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor := mul(8, 15)

Mobius inversion ORIGINAL inverse F(n/1) f(15) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor := mul(8, 18)

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor := mul(8, 30)

Mobius inversion ORIGINAL inverse F(n/1) f(30) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(30) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/1) f(45) ORIGINAL a times divisor := mul(8, 45)

Mobius inversion ORIGINAL inverse F(n/1) f(45) ORIGINAL a times divisor = 360

Mobius inversion ORIGINAL inverse F(n/1) f(45) plus ORIGINAL b := sum(360, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(45) plus ORIGINAL b = 361

Mobius inversion ORIGINAL inverse F(n/1) f(90) ORIGINAL a times divisor := mul(8, 90)

Mobius inversion ORIGINAL inverse F(n/1) f(90) ORIGINAL a times divisor = 720

Mobius inversion ORIGINAL inverse F(n/1) f(90) plus ORIGINAL b := sum(720, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(90) plus ORIGINAL b = 721

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(9, 17, 25, 41, 49, 73, 81, 121, 145, 241, 361, 721)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 1884

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 1884)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 1884

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(90, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 45

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=45;check complete integer range1..45

ALL positive divisors INCREASING=[1, 3, 5, 9, 15, 45];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(8, 1)

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

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor := mul(8, 9)

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor := mul(8, 15)

Mobius inversion ORIGINAL inverse F(n/2) f(15) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(15) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/2) f(45) ORIGINAL a times divisor := mul(8, 45)

Mobius inversion ORIGINAL inverse F(n/2) f(45) ORIGINAL a times divisor = 360

Mobius inversion ORIGINAL inverse F(n/2) f(45) plus ORIGINAL b := sum(360, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(45) plus ORIGINAL b = 361

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(9, 25, 41, 73, 121, 361)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 630

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 630)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -630

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(90, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 30

Mobius inversion ORIGINAL inverse F(n/3):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/3) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(8, 2)

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

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/3) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor := mul(8, 6)

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

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

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

Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor := mul(8, 10)

Mobius inversion ORIGINAL inverse F(n/3) f(10) ORIGINAL a times divisor = 80

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b := sum(80, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b = 81

Mobius inversion ORIGINAL inverse F(n/3) f(15) ORIGINAL a times divisor := mul(8, 15)

Mobius inversion ORIGINAL inverse F(n/3) f(15) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/3) f(15) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(15) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/3) f(30) ORIGINAL a times divisor := mul(8, 30)

Mobius inversion ORIGINAL inverse F(n/3) f(30) ORIGINAL a times divisor = 240

Mobius inversion ORIGINAL inverse F(n/3) f(30) plus ORIGINAL b := sum(240, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(30) plus ORIGINAL b = 241

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(9, 17, 25, 41, 49, 81, 121, 241)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 584

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 584)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -584

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(90, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 18

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=18;check complete integer range1..18

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

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(8, 2)

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/5) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor := mul(8, 6)

Mobius inversion ORIGINAL inverse F(n/5) f(6) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/5) f(9) ORIGINAL a times divisor := mul(8, 9)

Mobius inversion ORIGINAL inverse F(n/5) f(9) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/5) f(9) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(9) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/5) f(18) ORIGINAL a times divisor := mul(8, 18)

Mobius inversion ORIGINAL inverse F(n/5) f(18) ORIGINAL a times divisor = 144

Mobius inversion ORIGINAL inverse F(n/5) f(18) plus ORIGINAL b := sum(144, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(18) plus ORIGINAL b = 145

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(9, 17, 25, 49, 73, 145)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 318

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 318)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -318

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(90, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 15

Mobius inversion ORIGINAL inverse F(n/6):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/6) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/6) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/6) f(15) ORIGINAL a times divisor := mul(8, 15)

Mobius inversion ORIGINAL inverse F(n/6) f(15) ORIGINAL a times divisor = 120

Mobius inversion ORIGINAL inverse F(n/6) f(15) plus ORIGINAL b := sum(120, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(15) plus ORIGINAL b = 121

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(9, 25, 41, 121)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 196

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 196)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 196

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

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

ORIGINAL positive integer absolute magnitude=9>1

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

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

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

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

upf extract divisor3 occurrence0 = 3

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

upf extract divisor3 occurrence1 = 1

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

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

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 9

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 := div(90, 9)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 = 10

Mobius inversion ORIGINAL inverse F(n/9):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/9) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b := sum(8, 1)

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

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor := mul(8, 2)

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

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/9) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/9) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/9) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/9) f(10) ORIGINAL a times divisor := mul(8, 10)

Mobius inversion ORIGINAL inverse F(n/9) f(10) ORIGINAL a times divisor = 80

Mobius inversion ORIGINAL inverse F(n/9) f(10) plus ORIGINAL b := sum(80, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(10) plus ORIGINAL b = 81

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum := sum(9, 17, 41, 81)

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum = 148

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) := mul(0, 148)

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

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(90, 10)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 9

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=9;check complete integer range1..9

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

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/10) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/10) f(9) ORIGINAL a times divisor := mul(8, 9)

Mobius inversion ORIGINAL inverse F(n/10) f(9) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/10) f(9) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(9) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(9, 25, 73)

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 107

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 107)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 107

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(90, 15)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 = 6

Mobius inversion ORIGINAL inverse F(n/15):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/15) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor := mul(8, 2)

Mobius inversion ORIGINAL inverse F(n/15) f(2) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/15) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/15) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/15) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/15) f(6) ORIGINAL a times divisor := mul(8, 6)

Mobius inversion ORIGINAL inverse F(n/15) f(6) ORIGINAL a times divisor = 48

Mobius inversion ORIGINAL inverse F(n/15) f(6) plus ORIGINAL b := sum(48, 1)

Mobius inversion ORIGINAL inverse F(n/15) f(6) plus ORIGINAL b = 49

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum := sum(9, 17, 25, 49)

Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum = 100

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) := mul(1, 100)

Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) = 100

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

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

ORIGINAL positive integer absolute magnitude=18>1

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

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

upf extract divisor2 occurrence0 = 9

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

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

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

upf extract divisor3 occurrence1 = 3

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

upf extract divisor3 occurrence2 = 1

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

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 := div(90, 18)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 = 5

Mobius inversion ORIGINAL inverse F(n/18):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/18) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b := sum(8, 1)

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

Mobius inversion ORIGINAL inverse F(n/18) f(5) ORIGINAL a times divisor := mul(8, 5)

Mobius inversion ORIGINAL inverse F(n/18) f(5) ORIGINAL a times divisor = 40

Mobius inversion ORIGINAL inverse F(n/18) f(5) plus ORIGINAL b := sum(40, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(5) plus ORIGINAL b = 41

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum := sum(9, 41)

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum = 50

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) := mul(0, 50)

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

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(90, 30)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 = 3

Mobius inversion ORIGINAL inverse F(n/30):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/30) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/30) f(3) ORIGINAL a times divisor := mul(8, 3)

Mobius inversion ORIGINAL inverse F(n/30) f(3) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/30) f(3) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/30) f(3) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum := sum(9, 25)

Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum = 34

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) := mul(-1, 34)

Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) = -34

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor45:actual positive input for mu=45

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

ORIGINAL positive integer absolute magnitude=45>1

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

next divisor=2;previous remaining=45;all smaller divisors exhausted

extracted multiplicity=0;next remaining=45;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(45, 3)

upf extract divisor3 occurrence0 = 15

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

upf extract divisor3 occurrence1 = 5

next divisor=3;previous remaining=45;all smaller divisors exhausted

extracted multiplicity=2;next remaining=5;remaining modulo3=2;extracted multiset=[3, 3]

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

upf canonical complete product factor0 = 3

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

upf canonical complete product factor1 = 9

upf canonical complete product factor2 := mul(9, 5)

upf canonical complete product factor2 = 45

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=45

complete increasing prime multiset=[3, 3, 5];full product=45;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor45:input=45;COMPLETE prime multiset=[3, 3, 5];distinct primes=[3, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor45 := div(90, 45)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor45 = 2

Mobius inversion ORIGINAL inverse F(n/45):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/45) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/45) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/45) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/45) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/45) f(2) ORIGINAL a times divisor := mul(8, 2)

Mobius inversion ORIGINAL inverse F(n/45) f(2) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL inverse F(n/45) f(2) plus ORIGINAL b := sum(16, 1)

Mobius inversion ORIGINAL inverse F(n/45) f(2) plus ORIGINAL b = 17

Mobius inversion ORIGINAL inverse F(n/45) full forward divisor sum := sum(9, 17)

Mobius inversion ORIGINAL inverse F(n/45) full forward divisor sum = 26

Mobius inversion ORIGINAL inverse mu(45) times SAME F(n/45) := mul(0, 26)

Mobius inversion ORIGINAL inverse mu(45) times SAME F(n/45) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor90:actual positive input for mu=90

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

ORIGINAL positive integer absolute magnitude=90>1

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

upf extract divisor2 occurrence0 := div(90, 2)

upf extract divisor2 occurrence0 = 45

next divisor=2;previous remaining=90;all smaller divisors exhausted

extracted multiplicity=1;next remaining=45;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(45, 3)

upf extract divisor3 occurrence1 = 15

upf extract divisor3 occurrence2 := div(15, 3)

upf extract divisor3 occurrence2 = 5

next divisor=3;previous remaining=45;all smaller divisors exhausted

extracted multiplicity=2;next remaining=5;remaining modulo3=2;extracted multiset=[2, 3, 3]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

upf canonical complete product factor3 := mul(18, 5)

upf canonical complete product factor3 = 90

exhausted trial divisors below4;terminal remaining=5;d*d>remaining;ORIGINAL n=90

complete increasing prime multiset=[2, 3, 3, 5];full product=90;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor90:input=90;COMPLETE prime multiset=[2, 3, 3, 5];distinct primes=[2, 3, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor90 := div(90, 90)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor90 = 1

Mobius inversion ORIGINAL inverse F(n/90):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/90) f(1) ORIGINAL a times divisor := mul(8, 1)

Mobius inversion ORIGINAL inverse F(n/90) f(1) ORIGINAL a times divisor = 8

Mobius inversion ORIGINAL inverse F(n/90) f(1) plus ORIGINAL b := sum(8, 1)

Mobius inversion ORIGINAL inverse F(n/90) f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL inverse F(n/90) full forward divisor sum := sum(9)

Mobius inversion ORIGINAL inverse F(n/90) full forward divisor sum = 9

Mobius inversion ORIGINAL inverse mu(90) times SAME F(n/90) := mul(0, 9)

Mobius inversion ORIGINAL inverse mu(90) times SAME F(n/90) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(1884, -630, -584, -318, 196, 0, 107, 100, 0, -34, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 721

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(8, 90)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 720

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(720, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 721

Mobius inversion ORIGINAL inverse:actual n=90;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 90, 1884, 1884], [2, -1, 45, 630, -630], [3, -1, 30, 584, -584], [5, -1, 18, 318, -318], [6, 1, 15, 196, 196], [9, 0, 10, 148, 0], [10, 1, 9, 107, 107], [15, 1, 6, 100, 100], [18, 0, 5, 50, 0], [30, -1, 3, 34, -34], [45, 0, 2, 26, 0], [90, 0, 1, 9, 0]]

complete reconstructed=721;direct f(n)=721;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(-630, -584, -318, 196, 0, 107, 100, 0, -34, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -1163

Mobius inversion complete minus partial := sub(721, -1163)

Mobius inversion complete minus partial = 1884

ORIGINAL full divisors=[1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90]

complete=721=f(n)=721;partial=-1163;lost term=1884!=0;incomplete_certificate fixed No

Reverse checks

• 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 — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S261 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S260 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(90) times SAME F(n/90) with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/90) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/90) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/90) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor90 with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S251 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S250 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• 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 extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(45) times SAME F(n/45) with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/45) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/45) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/45) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/45) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/45) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor45 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S229 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S228 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(30) times SAME F(n/30) with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/30) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S215 · 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.
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor30 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 upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• 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 occurrence1 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 — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S194 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S187 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S184 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(15) times SAME F(n/15) with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S174 · 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.
• S173 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/15) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S172 · 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.
• 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor15 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S155 · 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.
• S154 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S153 · 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.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S115 · 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.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S111 · 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.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S093 · 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.
• S092 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S091 · 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.
• S090 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S089 · 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.
• S088 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S087 · 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.
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 canonical complete product factor0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(30) 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(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(15) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S071 · 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.
• S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S067 · 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.
• S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(45) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(45) 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(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · 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.
• S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · 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.
• S043 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(90) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(90) 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(45) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(45) 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(30) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · 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.
• S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(15) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · 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.
• 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(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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

(90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, 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.