第 22134 題：有限因數格上的Möbius反演

原題與已知條件

positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":110,"a":9,"b":1,"omitted_divisor":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={"n":110,"a":9,"b":1,"omitted_divisor":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-mobius-inversion-finite-divisors

逐步推導

source operation=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]

Mobius inversion ORIGINAL inverse:actual positive input=110;check complete integer range1..110

ALL positive divisors INCREASING=[1, 2, 5, 10, 11, 22, 55, 110];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

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

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

mu=1;empty product case=是;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(110, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 110

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=110;check complete integer range1..110

ALL positive divisors INCREASING=[1, 2, 5, 10, 11, 22, 55, 110];not only primes/proper divisors or all integers<=n

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(9, 2)

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(9, 5)

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 45

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(45, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 46

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(9, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(90, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(9, 11)

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 99

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(99, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 100

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(9, 22)

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 198

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(198, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 199

Mobius inversion ORIGINAL inverse F(n/1) f(55) ORIGINAL a times divisor := mul(9, 55)

Mobius inversion ORIGINAL inverse F(n/1) f(55) ORIGINAL a times divisor = 495

Mobius inversion ORIGINAL inverse F(n/1) f(55) plus ORIGINAL b := sum(495, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(55) plus ORIGINAL b = 496

Mobius inversion ORIGINAL inverse F(n/1) f(110) ORIGINAL a times divisor := mul(9, 110)

Mobius inversion ORIGINAL inverse F(n/1) f(110) ORIGINAL a times divisor = 990

Mobius inversion ORIGINAL inverse F(n/1) f(110) plus ORIGINAL b := sum(990, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(110) plus ORIGINAL b = 991

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(10, 19, 46, 91, 100, 199, 496, 991)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 1952

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 1952)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 1952

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

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

ORIGINAL positive integer absolute magnitude=2>1

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

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

upf canonical complete product factor0 = 2

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

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

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

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(110, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 55

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=55;check complete integer range1..55

ALL positive divisors INCREASING=[1, 5, 11, 55];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 9

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

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

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(9, 5)

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 45

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(45, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 46

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(9, 11)

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 99

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(99, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 100

Mobius inversion ORIGINAL inverse F(n/2) f(55) ORIGINAL a times divisor := mul(9, 55)

Mobius inversion ORIGINAL inverse F(n/2) f(55) ORIGINAL a times divisor = 495

Mobius inversion ORIGINAL inverse F(n/2) f(55) plus ORIGINAL b := sum(495, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(55) plus ORIGINAL b = 496

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(10, 46, 100, 496)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 652

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 652)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -652

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

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

ORIGINAL positive integer absolute magnitude=5>1

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

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

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

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

upf canonical complete product factor0 = 5

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

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

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

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(110, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 22

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=22;check complete integer range1..22

ALL positive divisors INCREASING=[1, 2, 11, 22];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 9

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

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

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(9, 2)

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

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

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/5) f(11) ORIGINAL a times divisor := mul(9, 11)

Mobius inversion ORIGINAL inverse F(n/5) f(11) ORIGINAL a times divisor = 99

Mobius inversion ORIGINAL inverse F(n/5) f(11) plus ORIGINAL b := sum(99, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(11) plus ORIGINAL b = 100

Mobius inversion ORIGINAL inverse F(n/5) f(22) ORIGINAL a times divisor := mul(9, 22)

Mobius inversion ORIGINAL inverse F(n/5) f(22) ORIGINAL a times divisor = 198

Mobius inversion ORIGINAL inverse F(n/5) f(22) plus ORIGINAL b := sum(198, 1)

Mobius inversion ORIGINAL inverse F(n/5) f(22) plus ORIGINAL b = 199

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(10, 19, 100, 199)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 328

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 328)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -328

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

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

ORIGINAL positive integer absolute magnitude=10>1

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

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

upf extract divisor2 occurrence0 = 5

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

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 10

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

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

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

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 := div(110, 10)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 11

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=11;check complete integer range1..11

ALL positive divisors INCREASING=[1, 11];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 9

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

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

Mobius inversion ORIGINAL inverse F(n/10) f(11) ORIGINAL a times divisor := mul(9, 11)

Mobius inversion ORIGINAL inverse F(n/10) f(11) ORIGINAL a times divisor = 99

Mobius inversion ORIGINAL inverse F(n/10) f(11) plus ORIGINAL b := sum(99, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(11) plus ORIGINAL b = 100

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(10, 100)

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 110

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 110)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 110

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:actual positive input for mu=11

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

ORIGINAL positive integer absolute magnitude=11>1

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

next divisor=2;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 11)

upf canonical complete product factor0 = 11

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=11

complete increasing prime multiset=[11];full product=11;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:input=11;COMPLETE prime multiset=[11];distinct primes=[11]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 := div(110, 11)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 10

Mobius inversion ORIGINAL inverse F(n/11):actual positive input=10;check complete integer range1..10

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

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 9

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(9, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 10

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(9, 2)

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/11) f(5) ORIGINAL a times divisor := mul(9, 5)

Mobius inversion ORIGINAL inverse F(n/11) f(5) ORIGINAL a times divisor = 45

Mobius inversion ORIGINAL inverse F(n/11) f(5) plus ORIGINAL b := sum(45, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(5) plus ORIGINAL b = 46

Mobius inversion ORIGINAL inverse F(n/11) f(10) ORIGINAL a times divisor := mul(9, 10)

Mobius inversion ORIGINAL inverse F(n/11) f(10) ORIGINAL a times divisor = 90

Mobius inversion ORIGINAL inverse F(n/11) f(10) plus ORIGINAL b := sum(90, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(10) plus ORIGINAL b = 91

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(10, 19, 46, 91)

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 166

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 166)

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -166

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:actual positive input for mu=22

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

ORIGINAL positive integer absolute magnitude=22>1

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

upf extract divisor2 occurrence0 := div(22, 2)

upf extract divisor2 occurrence0 = 11

next divisor=2;previous remaining=22;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=11;all smaller divisors exhausted

extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 11)

upf canonical complete product factor1 = 22

exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=22

complete increasing prime multiset=[2, 11];full product=22;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:input=22;COMPLETE prime multiset=[2, 11];distinct primes=[2, 11]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 := div(110, 22)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 5

Mobius inversion ORIGINAL inverse F(n/22):actual positive input=5;check complete integer range1..5

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

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 9

Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(9, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b = 10

Mobius inversion ORIGINAL inverse F(n/22) f(5) ORIGINAL a times divisor := mul(9, 5)

Mobius inversion ORIGINAL inverse F(n/22) f(5) ORIGINAL a times divisor = 45

Mobius inversion ORIGINAL inverse F(n/22) f(5) plus ORIGINAL b := sum(45, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(5) plus ORIGINAL b = 46

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(10, 46)

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 56

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 56)

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 56

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor55:actual positive input for mu=55

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

ORIGINAL positive integer absolute magnitude=55>1

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

next divisor=2;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=0;next remaining=55;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=0;next remaining=55;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=0;next remaining=55;remaining modulo4=3;extracted multiset=[]

upf extract divisor5 occurrence0 := div(55, 5)

upf extract divisor5 occurrence0 = 11

next divisor=5;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo5=1;extracted multiset=[5]

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

upf canonical complete product factor0 = 5

upf canonical complete product factor1 := mul(5, 11)

upf canonical complete product factor1 = 55

exhausted trial divisors below6;terminal remaining=11;d*d>remaining;ORIGINAL n=55

complete increasing prime multiset=[5, 11];full product=55;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor55:input=55;COMPLETE prime multiset=[5, 11];distinct primes=[5, 11]

mu=1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor55 := div(110, 55)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor55 = 2

Mobius inversion ORIGINAL inverse F(n/55):actual positive input=2;check complete integer range1..2

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

Mobius inversion ORIGINAL inverse F(n/55) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/55) f(1) ORIGINAL a times divisor = 9

Mobius inversion ORIGINAL inverse F(n/55) f(1) plus ORIGINAL b := sum(9, 1)

Mobius inversion ORIGINAL inverse F(n/55) f(1) plus ORIGINAL b = 10

Mobius inversion ORIGINAL inverse F(n/55) f(2) ORIGINAL a times divisor := mul(9, 2)

Mobius inversion ORIGINAL inverse F(n/55) f(2) ORIGINAL a times divisor = 18

Mobius inversion ORIGINAL inverse F(n/55) f(2) plus ORIGINAL b := sum(18, 1)

Mobius inversion ORIGINAL inverse F(n/55) f(2) plus ORIGINAL b = 19

Mobius inversion ORIGINAL inverse F(n/55) full forward divisor sum := sum(10, 19)

Mobius inversion ORIGINAL inverse F(n/55) full forward divisor sum = 29

Mobius inversion ORIGINAL inverse mu(55) times SAME F(n/55) := mul(1, 29)

Mobius inversion ORIGINAL inverse mu(55) times SAME F(n/55) = 29

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor110:actual positive input for mu=110

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

ORIGINAL positive integer absolute magnitude=110>1

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

upf extract divisor2 occurrence0 := div(110, 2)

upf extract divisor2 occurrence0 = 55

next divisor=2;previous remaining=110;all smaller divisors exhausted

extracted multiplicity=1;next remaining=55;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=0;next remaining=55;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=0;next remaining=55;remaining modulo4=3;extracted multiset=[2]

upf extract divisor5 occurrence1 := div(55, 5)

upf extract divisor5 occurrence1 = 11

next divisor=5;previous remaining=55;all smaller divisors exhausted

extracted multiplicity=1;next remaining=11;remaining modulo5=1;extracted multiset=[2, 5]

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 10

upf canonical complete product factor2 := mul(10, 11)

upf canonical complete product factor2 = 110

exhausted trial divisors below6;terminal remaining=11;d*d>remaining;ORIGINAL n=110

complete increasing prime multiset=[2, 5, 11];full product=110;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor110:input=110;COMPLETE prime multiset=[2, 5, 11];distinct primes=[2, 5, 11]

mu=-1;empty product case=否;repeated prime=否

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor110 := div(110, 110)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor110 = 1

Mobius inversion ORIGINAL inverse F(n/110):actual positive input=1;check complete integer range1..1

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

Mobius inversion ORIGINAL inverse F(n/110) f(1) ORIGINAL a times divisor := mul(9, 1)

Mobius inversion ORIGINAL inverse F(n/110) f(1) ORIGINAL a times divisor = 9

Mobius inversion ORIGINAL inverse F(n/110) f(1) plus ORIGINAL b := sum(9, 1)

Mobius inversion ORIGINAL inverse F(n/110) f(1) plus ORIGINAL b = 10

Mobius inversion ORIGINAL inverse F(n/110) full forward divisor sum := sum(10)

Mobius inversion ORIGINAL inverse F(n/110) full forward divisor sum = 10

Mobius inversion ORIGINAL inverse mu(110) times SAME F(n/110) := mul(-1, 10)

Mobius inversion ORIGINAL inverse mu(110) times SAME F(n/110) = -10

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(1952, -652, -328, 110, -166, 56, 29, -10)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 991

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(9, 110)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 990

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(990, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 991

Mobius inversion ORIGINAL inverse:actual n=110;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 110, 1952, 1952], [2, -1, 55, 652, -652], [5, -1, 22, 328, -328], [10, 1, 11, 110, 110], [11, -1, 10, 166, -166], [22, 1, 5, 56, 56], [55, 1, 2, 29, 29], [110, -1, 1, 10, -10]]

complete reconstructed=991;direct f(n)=991;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-652, -328, 110, -166, 56, 29, -10)

Mobius inversion ORIGINAL omitted-divisor partial sum = -961

Mobius inversion complete minus partial := sub(991, -961)

Mobius inversion complete minus partial = 1952

ORIGINAL full divisors=[1, 2, 5, 10, 11, 22, 55, 110];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 5, 10, 11, 22, 55, 110]

complete=991=f(n)=991;partial=-961;lost term=1952!=0;incomplete_certificate fixed No

逐步反向核對

• S156 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S155 · premise_and_result_audit — 以精確有理數重算 Mobius inversion complete minus partial，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL omitted-divisor partial sum，不以小數近似替代。
• S153 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S152 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct plus ORIGINAL b，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct ORIGINAL a times n，不以小數近似替代。
• S150 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse complete inverse-term sum，不以小數近似替代。
• S149 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(110) times SAME F(n/110)，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/110) full forward divisor sum，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/110) f(1) plus ORIGINAL b，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/110) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S145 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S144 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor110，不以小數近似替代。
• S143 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S142 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S141 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S138 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S137 · premise_and_result_audit — 以精確有理數重算 upf extract divisor5 occurrence1，不以小數近似替代。
• S136 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S135 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S134 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S133 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S132 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S131 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S130 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(55) times SAME F(n/55)，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/55) full forward divisor sum，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/55) f(2) plus ORIGINAL b，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/55) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/55) f(1) plus ORIGINAL b，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/55) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S124 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S123 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor55，不以小數近似替代。
• S122 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S121 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S120 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S118 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S117 · premise_and_result_audit — 以精確有理數重算 upf extract divisor5 occurrence0，不以小數近似替代。
• S116 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S115 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S114 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S113 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S112 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S111 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22)，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(5) plus ORIGINAL b，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S105 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S104 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22，不以小數近似替代。
• S103 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S102 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S101 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S099 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S098 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S097 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S096 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S095 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S094 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11)，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(10) plus ORIGINAL b，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(10) ORIGINAL a times divisor，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(5) plus ORIGINAL b，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b，不以小數近似替代。
• S087 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S084 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11，不以小數近似替代。
• S082 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S081 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S080 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S079 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S077 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S076 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S075 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10)，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/10) f(11) plus ORIGINAL b，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/10) f(11) ORIGINAL a times divisor，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S068 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10，不以小數近似替代。
• S067 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S066 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S065 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5)，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(22) plus ORIGINAL b，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(22) ORIGINAL a times divisor，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(11) plus ORIGINAL b，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(11) ORIGINAL a times divisor，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5，不以小數近似替代。
• S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S046 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S045 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(55) plus ORIGINAL b，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(55) ORIGINAL a times divisor，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，不以小數近似替代。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(110) plus ORIGINAL b，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(110) ORIGINAL a times divisor，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(55) plus ORIGINAL b，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(55) ORIGINAL a times divisor，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。