Question 22143: An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Original question and givens

Use positive integer n=380, exact integer arithmetic function f(k)=a*k+b with a=18, 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=380, exact integer arithmetic function f(k)=a*k+b with a=18, 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=380;check complete integer range1..380

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380];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(380, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 380

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=380;check complete integer range1..380

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380];not only primes/proper divisors or all integers<=n

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(18, 4)

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(90, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(18, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 180

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(180, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 181

Mobius inversion ORIGINAL inverse F(n/1) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/1) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/1) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor := mul(18, 20)

Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor = 360

Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b := sum(360, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b = 361

Mobius inversion ORIGINAL inverse F(n/1) f(38) ORIGINAL a times divisor := mul(18, 38)

Mobius inversion ORIGINAL inverse F(n/1) f(38) ORIGINAL a times divisor = 684

Mobius inversion ORIGINAL inverse F(n/1) f(38) plus ORIGINAL b := sum(684, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(38) plus ORIGINAL b = 685

Mobius inversion ORIGINAL inverse F(n/1) f(76) ORIGINAL a times divisor := mul(18, 76)

Mobius inversion ORIGINAL inverse F(n/1) f(76) ORIGINAL a times divisor = 1368

Mobius inversion ORIGINAL inverse F(n/1) f(76) plus ORIGINAL b := sum(1368, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(76) plus ORIGINAL b = 1369

Mobius inversion ORIGINAL inverse F(n/1) f(95) ORIGINAL a times divisor := mul(18, 95)

Mobius inversion ORIGINAL inverse F(n/1) f(95) ORIGINAL a times divisor = 1710

Mobius inversion ORIGINAL inverse F(n/1) f(95) plus ORIGINAL b := sum(1710, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(95) plus ORIGINAL b = 1711

Mobius inversion ORIGINAL inverse F(n/1) f(190) ORIGINAL a times divisor := mul(18, 190)

Mobius inversion ORIGINAL inverse F(n/1) f(190) ORIGINAL a times divisor = 3420

Mobius inversion ORIGINAL inverse F(n/1) f(190) plus ORIGINAL b := sum(3420, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(190) plus ORIGINAL b = 3421

Mobius inversion ORIGINAL inverse F(n/1) f(380) ORIGINAL a times divisor := mul(18, 380)

Mobius inversion ORIGINAL inverse F(n/1) f(380) ORIGINAL a times divisor = 6840

Mobius inversion ORIGINAL inverse F(n/1) f(380) plus ORIGINAL b := sum(6840, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(380) plus ORIGINAL b = 6841

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(19, 37, 73, 91, 181, 343, 361, 685, 1369, 1711, 3421, 6841)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 15132

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 15132)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 15132

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(380, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 190

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=190;check complete integer range1..190

ALL positive divisors INCREASING=[1, 2, 5, 10, 19, 38, 95, 190];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor := mul(18, 10)

Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor = 180

Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b := sum(180, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b = 181

Mobius inversion ORIGINAL inverse F(n/2) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/2) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/2) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/2) f(38) ORIGINAL a times divisor := mul(18, 38)

Mobius inversion ORIGINAL inverse F(n/2) f(38) ORIGINAL a times divisor = 684

Mobius inversion ORIGINAL inverse F(n/2) f(38) plus ORIGINAL b := sum(684, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(38) plus ORIGINAL b = 685

Mobius inversion ORIGINAL inverse F(n/2) f(95) ORIGINAL a times divisor := mul(18, 95)

Mobius inversion ORIGINAL inverse F(n/2) f(95) ORIGINAL a times divisor = 1710

Mobius inversion ORIGINAL inverse F(n/2) f(95) plus ORIGINAL b := sum(1710, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(95) plus ORIGINAL b = 1711

Mobius inversion ORIGINAL inverse F(n/2) f(190) ORIGINAL a times divisor := mul(18, 190)

Mobius inversion ORIGINAL inverse F(n/2) f(190) ORIGINAL a times divisor = 3420

Mobius inversion ORIGINAL inverse F(n/2) f(190) plus ORIGINAL b := sum(3420, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(190) plus ORIGINAL b = 3421

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(19, 37, 91, 181, 343, 685, 1711, 3421)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 6488

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 6488)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -6488

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

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

ORIGINAL positive integer absolute magnitude=4>1

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

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(380, 4)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 95

Mobius inversion ORIGINAL inverse F(n/4):actual positive input=95;check complete integer range1..95

ALL positive divisors INCREASING=[1, 5, 19, 95];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/4) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/4) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/4) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/4) f(95) ORIGINAL a times divisor := mul(18, 95)

Mobius inversion ORIGINAL inverse F(n/4) f(95) ORIGINAL a times divisor = 1710

Mobius inversion ORIGINAL inverse F(n/4) f(95) plus ORIGINAL b := sum(1710, 1)

Mobius inversion ORIGINAL inverse F(n/4) f(95) plus ORIGINAL b = 1711

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(19, 91, 343, 1711)

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 2164

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 2164)

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0

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(380, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 76

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=76;check complete integer range1..76

ALL positive divisors INCREASING=[1, 2, 4, 19, 38, 76];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 18

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

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor := mul(18, 4)

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/5) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/5) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/5) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/5) f(38) ORIGINAL a times divisor := mul(18, 38)

Mobius inversion ORIGINAL inverse F(n/5) f(38) ORIGINAL a times divisor = 684

Mobius inversion ORIGINAL inverse F(n/5) f(38) plus ORIGINAL b := sum(684, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(38) plus ORIGINAL b = 685

Mobius inversion ORIGINAL inverse F(n/5) f(76) ORIGINAL a times divisor := mul(18, 76)

Mobius inversion ORIGINAL inverse F(n/5) f(76) ORIGINAL a times divisor = 1368

Mobius inversion ORIGINAL inverse F(n/5) f(76) plus ORIGINAL b := sum(1368, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(76) plus ORIGINAL b = 1369

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(19, 37, 73, 343, 685, 1369)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 2526

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 2526)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -2526

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(380, 10)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 38

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=38;check complete integer range1..38

ALL positive divisors INCREASING=[1, 2, 19, 38];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/10) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/10) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/10) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/10) f(38) ORIGINAL a times divisor := mul(18, 38)

Mobius inversion ORIGINAL inverse F(n/10) f(38) ORIGINAL a times divisor = 684

Mobius inversion ORIGINAL inverse F(n/10) f(38) plus ORIGINAL b := sum(684, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(38) plus ORIGINAL b = 685

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(19, 37, 343, 685)

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 1084

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 1084)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 1084

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor19:actual positive input for mu=19

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

ORIGINAL positive integer absolute magnitude=19>1

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

next divisor=2;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 19)

upf canonical complete product factor0 = 19

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=19

complete increasing prime multiset=[19];full product=19;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor19:input=19;COMPLETE prime multiset=[19];distinct primes=[19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 := div(380, 19)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 = 20

Mobius inversion ORIGINAL inverse F(n/19):actual positive input=20;check complete integer range1..20

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 20];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/19) f(4) ORIGINAL a times divisor := mul(18, 4)

Mobius inversion ORIGINAL inverse F(n/19) f(4) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/19) f(4) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(4) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/19) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/19) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/19) f(5) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(5) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/19) f(10) ORIGINAL a times divisor := mul(18, 10)

Mobius inversion ORIGINAL inverse F(n/19) f(10) ORIGINAL a times divisor = 180

Mobius inversion ORIGINAL inverse F(n/19) f(10) plus ORIGINAL b := sum(180, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(10) plus ORIGINAL b = 181

Mobius inversion ORIGINAL inverse F(n/19) f(20) ORIGINAL a times divisor := mul(18, 20)

Mobius inversion ORIGINAL inverse F(n/19) f(20) ORIGINAL a times divisor = 360

Mobius inversion ORIGINAL inverse F(n/19) f(20) plus ORIGINAL b := sum(360, 1)

Mobius inversion ORIGINAL inverse F(n/19) f(20) plus ORIGINAL b = 361

Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum := sum(19, 37, 73, 91, 181, 361)

Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum = 762

Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) := mul(-1, 762)

Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) = -762

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:actual positive input for mu=20

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

ORIGINAL positive integer absolute magnitude=20>1

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

upf extract divisor2 occurrence0 := div(20, 2)

upf extract divisor2 occurrence0 = 10

upf extract divisor2 occurrence1 := div(10, 2)

upf extract divisor2 occurrence1 = 5

next divisor=2;previous remaining=20;all smaller divisors exhausted

extracted multiplicity=2;next remaining=5;remaining modulo2=1;extracted multiset=[2, 2]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 5)

upf canonical complete product factor2 = 20

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=20

complete increasing prime multiset=[2, 2, 5];full product=20;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:input=20;COMPLETE prime multiset=[2, 2, 5];distinct primes=[2, 5]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 := div(380, 20)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 = 19

Mobius inversion ORIGINAL inverse F(n/20):actual positive input=19;check complete integer range1..19

ALL positive divisors INCREASING=[1, 19];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/20) f(19) ORIGINAL a times divisor := mul(18, 19)

Mobius inversion ORIGINAL inverse F(n/20) f(19) ORIGINAL a times divisor = 342

Mobius inversion ORIGINAL inverse F(n/20) f(19) plus ORIGINAL b := sum(342, 1)

Mobius inversion ORIGINAL inverse F(n/20) f(19) plus ORIGINAL b = 343

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum := sum(19, 343)

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum = 362

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) := mul(0, 362)

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor38:actual positive input for mu=38

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

ORIGINAL positive integer absolute magnitude=38>1

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

upf extract divisor2 occurrence0 := div(38, 2)

upf extract divisor2 occurrence0 = 19

next divisor=2;previous remaining=38;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 19)

upf canonical complete product factor1 = 38

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=38

complete increasing prime multiset=[2, 19];full product=38;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor38:input=38;COMPLETE prime multiset=[2, 19];distinct primes=[2, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 := div(380, 38)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 = 10

Mobius inversion ORIGINAL inverse F(n/38):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/38) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/38) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/38) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/38) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/38) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/38) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/38) f(5) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(5) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/38) f(10) ORIGINAL a times divisor := mul(18, 10)

Mobius inversion ORIGINAL inverse F(n/38) f(10) ORIGINAL a times divisor = 180

Mobius inversion ORIGINAL inverse F(n/38) f(10) plus ORIGINAL b := sum(180, 1)

Mobius inversion ORIGINAL inverse F(n/38) f(10) plus ORIGINAL b = 181

Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum := sum(19, 37, 91, 181)

Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum = 328

Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) := mul(1, 328)

Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) = 328

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor76:actual positive input for mu=76

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

ORIGINAL positive integer absolute magnitude=76>1

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

upf extract divisor2 occurrence0 := div(76, 2)

upf extract divisor2 occurrence0 = 38

upf extract divisor2 occurrence1 := div(38, 2)

upf extract divisor2 occurrence1 = 19

next divisor=2;previous remaining=76;all smaller divisors exhausted

extracted multiplicity=2;next remaining=19;remaining modulo2=1;extracted multiset=[2, 2]

next divisor=3;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo3=1;extracted multiset=[2, 2]

next divisor=4;previous remaining=19;all smaller divisors exhausted

extracted multiplicity=0;next remaining=19;remaining modulo4=3;extracted multiset=[2, 2]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 19)

upf canonical complete product factor2 = 76

exhausted trial divisors below5;terminal remaining=19;d*d>remaining;ORIGINAL n=76

complete increasing prime multiset=[2, 2, 19];full product=76;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor76:input=76;COMPLETE prime multiset=[2, 2, 19];distinct primes=[2, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor76 := div(380, 76)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor76 = 5

Mobius inversion ORIGINAL inverse F(n/76):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/76) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/76) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/76) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/76) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/76) f(5) ORIGINAL a times divisor := mul(18, 5)

Mobius inversion ORIGINAL inverse F(n/76) f(5) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/76) f(5) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/76) f(5) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/76) full forward divisor sum := sum(19, 91)

Mobius inversion ORIGINAL inverse F(n/76) full forward divisor sum = 110

Mobius inversion ORIGINAL inverse mu(76) times SAME F(n/76) := mul(0, 110)

Mobius inversion ORIGINAL inverse mu(76) times SAME F(n/76) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor95:actual positive input for mu=95

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

ORIGINAL positive integer absolute magnitude=95>1

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

next divisor=2;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo4=3;extracted multiset=[]

upf extract divisor5 occurrence0 := div(95, 5)

upf extract divisor5 occurrence0 = 19

next divisor=5;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo5=4;extracted multiset=[5]

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

upf canonical complete product factor0 = 5

upf canonical complete product factor1 := mul(5, 19)

upf canonical complete product factor1 = 95

exhausted trial divisors below6;terminal remaining=19;d*d>remaining;ORIGINAL n=95

complete increasing prime multiset=[5, 19];full product=95;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor95:input=95;COMPLETE prime multiset=[5, 19];distinct primes=[5, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor95 := div(380, 95)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor95 = 4

Mobius inversion ORIGINAL inverse F(n/95):actual positive input=4;check complete integer range1..4

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/95) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/95) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/95) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/95) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/95) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/95) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/95) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/95) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/95) f(4) ORIGINAL a times divisor := mul(18, 4)

Mobius inversion ORIGINAL inverse F(n/95) f(4) ORIGINAL a times divisor = 72

Mobius inversion ORIGINAL inverse F(n/95) f(4) plus ORIGINAL b := sum(72, 1)

Mobius inversion ORIGINAL inverse F(n/95) f(4) plus ORIGINAL b = 73

Mobius inversion ORIGINAL inverse F(n/95) full forward divisor sum := sum(19, 37, 73)

Mobius inversion ORIGINAL inverse F(n/95) full forward divisor sum = 129

Mobius inversion ORIGINAL inverse mu(95) times SAME F(n/95) := mul(1, 129)

Mobius inversion ORIGINAL inverse mu(95) times SAME F(n/95) = 129

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor190:actual positive input for mu=190

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

ORIGINAL positive integer absolute magnitude=190>1

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

upf extract divisor2 occurrence0 := div(190, 2)

upf extract divisor2 occurrence0 = 95

next divisor=2;previous remaining=190;all smaller divisors exhausted

extracted multiplicity=1;next remaining=95;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo4=3;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(95, 5)

upf extract divisor5 occurrence1 = 19

next divisor=5;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo5=4;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, 19)

upf canonical complete product factor2 = 190

exhausted trial divisors below6;terminal remaining=19;d*d>remaining;ORIGINAL n=190

complete increasing prime multiset=[2, 5, 19];full product=190;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor190:input=190;COMPLETE prime multiset=[2, 5, 19];distinct primes=[2, 5, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor190 := div(380, 190)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor190 = 2

Mobius inversion ORIGINAL inverse F(n/190):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/190) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/190) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/190) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/190) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/190) f(2) ORIGINAL a times divisor := mul(18, 2)

Mobius inversion ORIGINAL inverse F(n/190) f(2) ORIGINAL a times divisor = 36

Mobius inversion ORIGINAL inverse F(n/190) f(2) plus ORIGINAL b := sum(36, 1)

Mobius inversion ORIGINAL inverse F(n/190) f(2) plus ORIGINAL b = 37

Mobius inversion ORIGINAL inverse F(n/190) full forward divisor sum := sum(19, 37)

Mobius inversion ORIGINAL inverse F(n/190) full forward divisor sum = 56

Mobius inversion ORIGINAL inverse mu(190) times SAME F(n/190) := mul(-1, 56)

Mobius inversion ORIGINAL inverse mu(190) times SAME F(n/190) = -56

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor380:actual positive input for mu=380

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

ORIGINAL positive integer absolute magnitude=380>1

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

upf extract divisor2 occurrence0 := div(380, 2)

upf extract divisor2 occurrence0 = 190

upf extract divisor2 occurrence1 := div(190, 2)

upf extract divisor2 occurrence1 = 95

next divisor=2;previous remaining=380;all smaller divisors exhausted

extracted multiplicity=2;next remaining=95;remaining modulo2=1;extracted multiset=[2, 2]

next divisor=3;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo3=2;extracted multiset=[2, 2]

next divisor=4;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=0;next remaining=95;remaining modulo4=3;extracted multiset=[2, 2]

upf extract divisor5 occurrence2 := div(95, 5)

upf extract divisor5 occurrence2 = 19

next divisor=5;previous remaining=95;all smaller divisors exhausted

extracted multiplicity=1;next remaining=19;remaining modulo5=4;extracted multiset=[2, 2, 5]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 5)

upf canonical complete product factor2 = 20

upf canonical complete product factor3 := mul(20, 19)

upf canonical complete product factor3 = 380

exhausted trial divisors below6;terminal remaining=19;d*d>remaining;ORIGINAL n=380

complete increasing prime multiset=[2, 2, 5, 19];full product=380;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor380:input=380;COMPLETE prime multiset=[2, 2, 5, 19];distinct primes=[2, 5, 19]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor380 := div(380, 380)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor380 = 1

Mobius inversion ORIGINAL inverse F(n/380):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/380) f(1) ORIGINAL a times divisor := mul(18, 1)

Mobius inversion ORIGINAL inverse F(n/380) f(1) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/380) f(1) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/380) f(1) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/380) full forward divisor sum := sum(19)

Mobius inversion ORIGINAL inverse F(n/380) full forward divisor sum = 19

Mobius inversion ORIGINAL inverse mu(380) times SAME F(n/380) := mul(0, 19)

Mobius inversion ORIGINAL inverse mu(380) times SAME F(n/380) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(15132, -6488, 0, -2526, 1084, -762, 0, 328, 0, 129, -56, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 6841

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(18, 380)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 6840

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(6840, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 6841

Mobius inversion ORIGINAL inverse:actual n=380;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 380, 15132, 15132], [2, -1, 190, 6488, -6488], [4, 0, 95, 2164, 0], [5, -1, 76, 2526, -2526], [10, 1, 38, 1084, 1084], [19, -1, 20, 762, -762], [20, 0, 19, 362, 0], [38, 1, 10, 328, 328], [76, 0, 5, 110, 0], [95, 1, 4, 129, 129], [190, -1, 2, 56, -56], [380, 0, 1, 19, 0]]

complete reconstructed=6841;direct f(n)=6841;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(-6488, 0, -2526, 1084, -762, 0, 328, 0, 129, -56, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -8291

Mobius inversion complete minus partial := sub(6841, -8291)

Mobius inversion complete minus partial = 15132

ORIGINAL full divisors=[1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380]

complete=6841=f(n)=6841;partial=-8291;lost term=15132!=0;incomplete_certificate fixed No

Reverse checks

• S274 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S273 · premise_and_result_audit — Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals.
• S272 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S270 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(380) times SAME F(n/380) with exact rational arithmetic, not rounded decimals.
• S266 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/380) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/380) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/380) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S262 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor380 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S259 · premise_and_result_audit — Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S254 · premise_and_result_audit — Recompute upf extract divisor5 occurrence2 with exact rational arithmetic, not rounded decimals.
• 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 — 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 extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S247 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S246 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(190) times SAME F(n/190) with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/190) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/190) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/190) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/190) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/190) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S239 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor190 with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S236 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S232 · premise_and_result_audit — Recompute upf extract divisor5 occurrence1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S228 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S225 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(95) times SAME F(n/95) with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/95) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S216 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor95 with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S214 · premise_and_result_audit — 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 canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S212 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S210 · premise_and_result_audit — Recompute upf extract divisor5 occurrence0 with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S204 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(76) times SAME F(n/76) with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/76) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/76) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/76) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/76) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/76) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor76 with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S194 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S184 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(38) times SAME F(n/38) with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/38) f(1) ORIGINAL a times divisor 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor38 with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 — 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 upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• 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 extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(19) times SAME F(n/19) with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) 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/19) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) 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/19) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/19) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor19 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(38) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S111 · 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.
• 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 divisor10 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(76) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(76) 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(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(38) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(19) 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(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) 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(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(95) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(95) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(19) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 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 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(190) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(190) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(95) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(95) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(38) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(19) 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(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) 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(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) 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(380) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(380) 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(190) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(190) 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(95) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(95) 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(76) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(76) 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(38) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(38) 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(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) 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(19) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(19) 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(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S015 · 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.
• 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(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) 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

(380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, 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.