Question 22193: Every Divisor Row Recovers Its Original Arithmetic Value

Original question and givens

Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. 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. Give (n, roundtrip_rows, all_equal, boundary).

• G001 — Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. 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. Give (n, roundtrip_rows, all_equal, 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-roundtrip-table;original snapshot and mathematics retained

explicit ordered source output fields=[n, roundtrip_rows, all_equal, boundary]

Mobius inversion ORIGINAL roundtrip: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 roundtrip m=1 forward:actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=1 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=1 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=1 forward full forward divisor sum := sum(8)

Mobius roundtrip m=1 forward full forward divisor sum = 8

Mobius roundtrip m=1 inverse:actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=1 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 roundtrip m=1 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 := div(1, 1)

Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 = 1

Mobius roundtrip m=1 inverse F(n/1):actual positive input=1;check complete integer range1..1

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

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

Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

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

Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum := sum(8)

Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum = 8

Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) := mul(1, 8)

Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) = 8

Mobius roundtrip m=1 inverse complete inverse-term sum := sum(8)

Mobius roundtrip m=1 inverse complete inverse-term sum = 8

Mobius roundtrip m=1 inverse direct ORIGINAL a times n := mul(3, 1)

Mobius roundtrip m=1 inverse direct ORIGINAL a times n = 3

Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=1 inverse direct plus ORIGINAL b = 8

Mobius roundtrip m=1 inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, 8, 8]]

complete reconstructed=8;direct f(n)=8;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=2 forward: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 roundtrip m=2 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=2 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=2 forward f(2) plus ORIGINAL b = 11

Mobius roundtrip m=2 forward full forward divisor sum := sum(8, 11)

Mobius roundtrip m=2 forward full forward divisor sum = 19

Mobius roundtrip m=2 inverse: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 roundtrip m=2 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 roundtrip m=2 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 := div(2, 1)

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 = 2

Mobius roundtrip m=2 inverse F(n/1):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 roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum := sum(8, 11)

Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum = 19

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) := mul(1, 19)

Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) = 19

Mobius roundtrip m=2 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 roundtrip m=2 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

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

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 := div(2, 2)

Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 = 1

Mobius roundtrip m=2 inverse F(n/2):actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum := sum(8)

Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum = 8

Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) := mul(-1, 8)

Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) = -8

Mobius roundtrip m=2 inverse complete inverse-term sum := sum(19, -8)

Mobius roundtrip m=2 inverse complete inverse-term sum = 11

Mobius roundtrip m=2 inverse direct ORIGINAL a times n := mul(3, 2)

Mobius roundtrip m=2 inverse direct ORIGINAL a times n = 6

Mobius roundtrip m=2 inverse direct plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=2 inverse direct plus ORIGINAL b = 11

Mobius roundtrip m=2 inverse:actual n=2;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 2, 19, 19], [2, -1, 1, 8, -8]]

complete reconstructed=11;direct f(n)=11;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=3 forward: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 roundtrip m=3 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=3 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=3 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=3 forward f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=3 forward f(3) plus ORIGINAL b = 14

Mobius roundtrip m=3 forward full forward divisor sum := sum(8, 14)

Mobius roundtrip m=3 forward full forward divisor sum = 22

Mobius roundtrip m=3 inverse: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 roundtrip m=3 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 roundtrip m=3 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 := div(3, 1)

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 = 3

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

Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

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

Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b = 8

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

Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b := sum(9, 5)

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

Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum := sum(8, 14)

Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum = 22

Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) := mul(1, 22)

Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) = 22

Mobius roundtrip m=3 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 roundtrip m=3 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 := div(3, 3)

Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 = 1

Mobius roundtrip m=3 inverse F(n/3):actual positive input=1;check complete integer range1..1

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

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

Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor = 3

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

Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum := sum(8)

Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum = 8

Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) := mul(-1, 8)

Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) = -8

Mobius roundtrip m=3 inverse complete inverse-term sum := sum(22, -8)

Mobius roundtrip m=3 inverse complete inverse-term sum = 14

Mobius roundtrip m=3 inverse direct ORIGINAL a times n := mul(3, 3)

Mobius roundtrip m=3 inverse direct ORIGINAL a times n = 9

Mobius roundtrip m=3 inverse direct plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=3 inverse direct plus ORIGINAL b = 14

Mobius roundtrip m=3 inverse:actual n=3;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 3, 22, 22], [3, -1, 1, 8, -8]]

complete reconstructed=14;direct f(n)=14;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=6 forward: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 roundtrip m=6 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=6 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=6 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=6 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=6 forward f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=6 forward f(2) plus ORIGINAL b = 11

Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=6 forward f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=6 forward f(3) plus ORIGINAL b = 14

Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor := mul(3, 6)

Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor = 18

Mobius roundtrip m=6 forward f(6) plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=6 forward f(6) plus ORIGINAL b = 23

Mobius roundtrip m=6 forward full forward divisor sum := sum(8, 11, 14, 23)

Mobius roundtrip m=6 forward full forward divisor sum = 56

Mobius roundtrip m=6 inverse: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 roundtrip m=6 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 roundtrip m=6 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1 := div(6, 1)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1 = 6

Mobius roundtrip m=6 inverse F(n/1):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 roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor := mul(3, 6)

Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor = 18

Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b = 23

Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum := sum(8, 11, 14, 23)

Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum = 56

Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1) := mul(1, 56)

Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1) = 56

Mobius roundtrip m=6 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 roundtrip m=6 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

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

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2 := div(6, 2)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2 = 3

Mobius roundtrip m=6 inverse F(n/2):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 roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum := sum(8, 14)

Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum = 22

Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2) := mul(-1, 22)

Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2) = -22

Mobius roundtrip m=6 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 roundtrip m=6 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3 := div(6, 3)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3 = 2

Mobius roundtrip m=6 inverse F(n/3):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 roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum := sum(8, 11)

Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum = 19

Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3) := mul(-1, 19)

Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3) = -19

Mobius roundtrip m=6 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 roundtrip m=6 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 roundtrip m=6 inverse n divided by ORIGINAL divisor6 := div(6, 6)

Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor6 = 1

Mobius roundtrip m=6 inverse F(n/6):actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum := sum(8)

Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum = 8

Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6) := mul(1, 8)

Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6) = 8

Mobius roundtrip m=6 inverse complete inverse-term sum := sum(56, -22, -19, 8)

Mobius roundtrip m=6 inverse complete inverse-term sum = 23

Mobius roundtrip m=6 inverse direct ORIGINAL a times n := mul(3, 6)

Mobius roundtrip m=6 inverse direct ORIGINAL a times n = 18

Mobius roundtrip m=6 inverse direct plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=6 inverse direct plus ORIGINAL b = 23

Mobius roundtrip m=6 inverse:actual n=6;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 6, 56, 56], [2, -1, 3, 22, -22], [3, -1, 2, 19, -19], [6, 1, 1, 8, 8]]

complete reconstructed=23;direct f(n)=23;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=9 forward: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 roundtrip m=9 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=9 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=9 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=9 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=9 forward f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=9 forward f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=9 forward f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=9 forward f(3) plus ORIGINAL b = 14

Mobius roundtrip m=9 forward f(9) ORIGINAL a times divisor := mul(3, 9)

Mobius roundtrip m=9 forward f(9) ORIGINAL a times divisor = 27

Mobius roundtrip m=9 forward f(9) plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=9 forward f(9) plus ORIGINAL b = 32

Mobius roundtrip m=9 forward full forward divisor sum := sum(8, 14, 32)

Mobius roundtrip m=9 forward full forward divisor sum = 54

Mobius roundtrip m=9 inverse: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 roundtrip m=9 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 roundtrip m=9 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor1 := div(9, 1)

Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor1 = 9

Mobius roundtrip m=9 inverse F(n/1):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 roundtrip m=9 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=9 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=9 inverse F(n/1) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=9 inverse F(n/1) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=9 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=9 inverse F(n/1) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=9 inverse F(n/1) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=9 inverse F(n/1) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=9 inverse F(n/1) f(9) ORIGINAL a times divisor := mul(3, 9)

Mobius roundtrip m=9 inverse F(n/1) f(9) ORIGINAL a times divisor = 27

Mobius roundtrip m=9 inverse F(n/1) f(9) plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=9 inverse F(n/1) f(9) plus ORIGINAL b = 32

Mobius roundtrip m=9 inverse F(n/1) full forward divisor sum := sum(8, 14, 32)

Mobius roundtrip m=9 inverse F(n/1) full forward divisor sum = 54

Mobius roundtrip m=9 inverse mu(1) times SAME F(n/1) := mul(1, 54)

Mobius roundtrip m=9 inverse mu(1) times SAME F(n/1) = 54

Mobius roundtrip m=9 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 roundtrip m=9 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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

Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor3 := div(9, 3)

Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor3 = 3

Mobius roundtrip m=9 inverse F(n/3):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 roundtrip m=9 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=9 inverse F(n/3) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=9 inverse F(n/3) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=9 inverse F(n/3) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=9 inverse F(n/3) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=9 inverse F(n/3) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=9 inverse F(n/3) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=9 inverse F(n/3) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=9 inverse F(n/3) full forward divisor sum := sum(8, 14)

Mobius roundtrip m=9 inverse F(n/3) full forward divisor sum = 22

Mobius roundtrip m=9 inverse mu(3) times SAME F(n/3) := mul(-1, 22)

Mobius roundtrip m=9 inverse mu(3) times SAME F(n/3) = -22

Mobius roundtrip m=9 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 roundtrip m=9 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 roundtrip m=9 inverse n divided by ORIGINAL divisor9 := div(9, 9)

Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor9 = 1

Mobius roundtrip m=9 inverse F(n/9):actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=9 inverse F(n/9) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=9 inverse F(n/9) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=9 inverse F(n/9) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=9 inverse F(n/9) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=9 inverse F(n/9) full forward divisor sum := sum(8)

Mobius roundtrip m=9 inverse F(n/9) full forward divisor sum = 8

Mobius roundtrip m=9 inverse mu(9) times SAME F(n/9) := mul(0, 8)

Mobius roundtrip m=9 inverse mu(9) times SAME F(n/9) = 0

Mobius roundtrip m=9 inverse complete inverse-term sum := sum(54, -22, 0)

Mobius roundtrip m=9 inverse complete inverse-term sum = 32

Mobius roundtrip m=9 inverse direct ORIGINAL a times n := mul(3, 9)

Mobius roundtrip m=9 inverse direct ORIGINAL a times n = 27

Mobius roundtrip m=9 inverse direct plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=9 inverse direct plus ORIGINAL b = 32

Mobius roundtrip m=9 inverse:actual n=9;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 9, 54, 54], [3, -1, 3, 22, -22], [9, 0, 1, 8, 0]]

complete reconstructed=32;direct f(n)=32;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius roundtrip m=18 forward: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 roundtrip m=18 forward f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 forward f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 forward f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 forward f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 forward f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=18 forward f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=18 forward f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=18 forward f(2) plus ORIGINAL b = 11

Mobius roundtrip m=18 forward f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=18 forward f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=18 forward f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=18 forward f(3) plus ORIGINAL b = 14

Mobius roundtrip m=18 forward f(6) ORIGINAL a times divisor := mul(3, 6)

Mobius roundtrip m=18 forward f(6) ORIGINAL a times divisor = 18

Mobius roundtrip m=18 forward f(6) plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=18 forward f(6) plus ORIGINAL b = 23

Mobius roundtrip m=18 forward f(9) ORIGINAL a times divisor := mul(3, 9)

Mobius roundtrip m=18 forward f(9) ORIGINAL a times divisor = 27

Mobius roundtrip m=18 forward f(9) plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=18 forward f(9) plus ORIGINAL b = 32

Mobius roundtrip m=18 forward f(18) ORIGINAL a times divisor := mul(3, 18)

Mobius roundtrip m=18 forward f(18) ORIGINAL a times divisor = 54

Mobius roundtrip m=18 forward f(18) plus ORIGINAL b := sum(54, 5)

Mobius roundtrip m=18 forward f(18) plus ORIGINAL b = 59

Mobius roundtrip m=18 forward full forward divisor sum := sum(8, 11, 14, 23, 32, 59)

Mobius roundtrip m=18 forward full forward divisor sum = 147

Mobius roundtrip m=18 inverse: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 roundtrip m=18 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 roundtrip m=18 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor1 := div(18, 1)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor1 = 18

Mobius roundtrip m=18 inverse F(n/1):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 roundtrip m=18 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/1) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/1) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/1) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/1) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=18 inverse F(n/1) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=18 inverse F(n/1) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=18 inverse F(n/1) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=18 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=18 inverse F(n/1) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=18 inverse F(n/1) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=18 inverse F(n/1) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=18 inverse F(n/1) f(6) ORIGINAL a times divisor := mul(3, 6)

Mobius roundtrip m=18 inverse F(n/1) f(6) ORIGINAL a times divisor = 18

Mobius roundtrip m=18 inverse F(n/1) f(6) plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=18 inverse F(n/1) f(6) plus ORIGINAL b = 23

Mobius roundtrip m=18 inverse F(n/1) f(9) ORIGINAL a times divisor := mul(3, 9)

Mobius roundtrip m=18 inverse F(n/1) f(9) ORIGINAL a times divisor = 27

Mobius roundtrip m=18 inverse F(n/1) f(9) plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=18 inverse F(n/1) f(9) plus ORIGINAL b = 32

Mobius roundtrip m=18 inverse F(n/1) f(18) ORIGINAL a times divisor := mul(3, 18)

Mobius roundtrip m=18 inverse F(n/1) f(18) ORIGINAL a times divisor = 54

Mobius roundtrip m=18 inverse F(n/1) f(18) plus ORIGINAL b := sum(54, 5)

Mobius roundtrip m=18 inverse F(n/1) f(18) plus ORIGINAL b = 59

Mobius roundtrip m=18 inverse F(n/1) full forward divisor sum := sum(8, 11, 14, 23, 32, 59)

Mobius roundtrip m=18 inverse F(n/1) full forward divisor sum = 147

Mobius roundtrip m=18 inverse mu(1) times SAME F(n/1) := mul(1, 147)

Mobius roundtrip m=18 inverse mu(1) times SAME F(n/1) = 147

Mobius roundtrip m=18 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 roundtrip m=18 inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

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

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor2 := div(18, 2)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor2 = 9

Mobius roundtrip m=18 inverse F(n/2):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 roundtrip m=18 inverse F(n/2) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/2) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/2) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/2) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/2) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=18 inverse F(n/2) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=18 inverse F(n/2) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=18 inverse F(n/2) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=18 inverse F(n/2) f(9) ORIGINAL a times divisor := mul(3, 9)

Mobius roundtrip m=18 inverse F(n/2) f(9) ORIGINAL a times divisor = 27

Mobius roundtrip m=18 inverse F(n/2) f(9) plus ORIGINAL b := sum(27, 5)

Mobius roundtrip m=18 inverse F(n/2) f(9) plus ORIGINAL b = 32

Mobius roundtrip m=18 inverse F(n/2) full forward divisor sum := sum(8, 14, 32)

Mobius roundtrip m=18 inverse F(n/2) full forward divisor sum = 54

Mobius roundtrip m=18 inverse mu(2) times SAME F(n/2) := mul(-1, 54)

Mobius roundtrip m=18 inverse mu(2) times SAME F(n/2) = -54

Mobius roundtrip m=18 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 roundtrip m=18 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor3 := div(18, 3)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor3 = 6

Mobius roundtrip m=18 inverse F(n/3):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 roundtrip m=18 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/3) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/3) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/3) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/3) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=18 inverse F(n/3) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=18 inverse F(n/3) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=18 inverse F(n/3) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=18 inverse F(n/3) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=18 inverse F(n/3) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=18 inverse F(n/3) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=18 inverse F(n/3) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=18 inverse F(n/3) f(6) ORIGINAL a times divisor := mul(3, 6)

Mobius roundtrip m=18 inverse F(n/3) f(6) ORIGINAL a times divisor = 18

Mobius roundtrip m=18 inverse F(n/3) f(6) plus ORIGINAL b := sum(18, 5)

Mobius roundtrip m=18 inverse F(n/3) f(6) plus ORIGINAL b = 23

Mobius roundtrip m=18 inverse F(n/3) full forward divisor sum := sum(8, 11, 14, 23)

Mobius roundtrip m=18 inverse F(n/3) full forward divisor sum = 56

Mobius roundtrip m=18 inverse mu(3) times SAME F(n/3) := mul(-1, 56)

Mobius roundtrip m=18 inverse mu(3) times SAME F(n/3) = -56

Mobius roundtrip m=18 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 roundtrip m=18 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 roundtrip m=18 inverse n divided by ORIGINAL divisor6 := div(18, 6)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor6 = 3

Mobius roundtrip m=18 inverse F(n/6):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 roundtrip m=18 inverse F(n/6) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/6) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/6) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/6) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/6) f(3) ORIGINAL a times divisor := mul(3, 3)

Mobius roundtrip m=18 inverse F(n/6) f(3) ORIGINAL a times divisor = 9

Mobius roundtrip m=18 inverse F(n/6) f(3) plus ORIGINAL b := sum(9, 5)

Mobius roundtrip m=18 inverse F(n/6) f(3) plus ORIGINAL b = 14

Mobius roundtrip m=18 inverse F(n/6) full forward divisor sum := sum(8, 14)

Mobius roundtrip m=18 inverse F(n/6) full forward divisor sum = 22

Mobius roundtrip m=18 inverse mu(6) times SAME F(n/6) := mul(1, 22)

Mobius roundtrip m=18 inverse mu(6) times SAME F(n/6) = 22

Mobius roundtrip m=18 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 roundtrip m=18 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 roundtrip m=18 inverse n divided by ORIGINAL divisor9 := div(18, 9)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor9 = 2

Mobius roundtrip m=18 inverse F(n/9):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 roundtrip m=18 inverse F(n/9) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/9) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/9) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/9) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/9) f(2) ORIGINAL a times divisor := mul(3, 2)

Mobius roundtrip m=18 inverse F(n/9) f(2) ORIGINAL a times divisor = 6

Mobius roundtrip m=18 inverse F(n/9) f(2) plus ORIGINAL b := sum(6, 5)

Mobius roundtrip m=18 inverse F(n/9) f(2) plus ORIGINAL b = 11

Mobius roundtrip m=18 inverse F(n/9) full forward divisor sum := sum(8, 11)

Mobius roundtrip m=18 inverse F(n/9) full forward divisor sum = 19

Mobius roundtrip m=18 inverse mu(9) times SAME F(n/9) := mul(0, 19)

Mobius roundtrip m=18 inverse mu(9) times SAME F(n/9) = 0

Mobius roundtrip m=18 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 roundtrip m=18 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 roundtrip m=18 inverse n divided by ORIGINAL divisor18 := div(18, 18)

Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor18 = 1

Mobius roundtrip m=18 inverse F(n/18):actual positive input=1;check complete integer range1..1

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

Mobius roundtrip m=18 inverse F(n/18) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius roundtrip m=18 inverse F(n/18) f(1) ORIGINAL a times divisor = 3

Mobius roundtrip m=18 inverse F(n/18) f(1) plus ORIGINAL b := sum(3, 5)

Mobius roundtrip m=18 inverse F(n/18) f(1) plus ORIGINAL b = 8

Mobius roundtrip m=18 inverse F(n/18) full forward divisor sum := sum(8)

Mobius roundtrip m=18 inverse F(n/18) full forward divisor sum = 8

Mobius roundtrip m=18 inverse mu(18) times SAME F(n/18) := mul(0, 8)

Mobius roundtrip m=18 inverse mu(18) times SAME F(n/18) = 0

Mobius roundtrip m=18 inverse complete inverse-term sum := sum(147, -54, -56, 22, 0, 0)

Mobius roundtrip m=18 inverse complete inverse-term sum = 59

Mobius roundtrip m=18 inverse direct ORIGINAL a times n := mul(3, 18)

Mobius roundtrip m=18 inverse direct ORIGINAL a times n = 54

Mobius roundtrip m=18 inverse direct plus ORIGINAL b := sum(54, 5)

Mobius roundtrip m=18 inverse direct plus ORIGINAL b = 59

Mobius roundtrip m=18 inverse:actual n=18;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]]

complete reconstructed=59;direct f(n)=59;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Reverse checks

• S327 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S326 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S325 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S324 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S323 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse mu(18) times SAME F(n/18) with exact rational arithmetic, not rounded decimals.
• S322 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/18) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S321 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/18) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S320 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/18) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S319 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S318 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor18 with exact rational arithmetic, not rounded decimals.
• S317 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S316 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S315 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
• S314 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S313 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S312 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S311 · premise_and_result_audit — Recompute upf extract divisor3 occurrence2 with exact rational arithmetic, not rounded decimals.
• S310 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S309 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S308 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S306 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S305 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse mu(9) times SAME F(n/9) with exact rational arithmetic, not rounded decimals.
• S304 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/9) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S303 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/9) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S302 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/9) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S301 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/9) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S300 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/9) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S299 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S298 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor9 with exact rational arithmetic, not rounded decimals.
• S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S296 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S295 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S294 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S293 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S292 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S291 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S290 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S289 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S288 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S287 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S286 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S285 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/6) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S284 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/6) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S280 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S278 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S277 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S276 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S275 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S274 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S272 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S271 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S266 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S260 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S258 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S257 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S254 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S253 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S250 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S245 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S242 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 roundtrip m=18 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S232 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S228 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S224 · premise_and_result_audit — Recompute Mobius roundtrip m=18 inverse n divided by ORIGINAL divisor1 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 roundtrip m=18 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(18) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(18) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S214 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S212 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S210 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute Mobius roundtrip m=18 forward f(1) ORIGINAL a times divisor 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S205 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse mu(9) times SAME F(n/9) with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/9) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/9) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/9) 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 roundtrip m=9 inverse n divided by ORIGINAL divisor9 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 factor1 with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S191 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S186 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/3) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/3) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S179 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S176 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 roundtrip m=9 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute Mobius roundtrip m=9 inverse F(n/1) f(1) ORIGINAL a times divisor 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 Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor1 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 roundtrip m=9 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(9) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(9) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Mobius roundtrip m=9 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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 — 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 roundtrip m=6 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/3) f(1) ORIGINAL a times divisor 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 — Recompute Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor3 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Recompute Mobius roundtrip m=6 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Recompute upf canonical complete product factor0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Recompute Mobius roundtrip m=6 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Mobius roundtrip m=6 forward f(1) ORIGINAL a times divisor 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 Mobius roundtrip m=3 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor 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 Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 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 — 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 roundtrip m=3 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Mobius roundtrip m=3 forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Mobius roundtrip m=3 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute Mobius roundtrip m=2 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute Mobius roundtrip m=2 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Mobius roundtrip m=2 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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

(18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], Yes, finite_divisor_poset_roundtrip)

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.