第 22137 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":182,"a":12,"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":182,"a":12,"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=182;check complete integer range1..182

ALL positive divisors INCREASING=[1, 2, 7, 13, 14, 26, 91, 182];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(182, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 182

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=182;check complete integer range1..182

ALL positive divisors INCREASING=[1, 2, 7, 13, 14, 26, 91, 182];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(12, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(12, 2)

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor := mul(12, 7)

Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 84

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(84, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 85

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(12, 13)

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 156

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(156, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 157

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor := mul(12, 14)

Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 168

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b := sum(168, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 169

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor := mul(12, 26)

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/1) f(91) ORIGINAL a times divisor := mul(12, 91)

Mobius inversion ORIGINAL inverse F(n/1) f(91) ORIGINAL a times divisor = 1092

Mobius inversion ORIGINAL inverse F(n/1) f(91) plus ORIGINAL b := sum(1092, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(91) plus ORIGINAL b = 1093

Mobius inversion ORIGINAL inverse F(n/1) f(182) ORIGINAL a times divisor := mul(12, 182)

Mobius inversion ORIGINAL inverse F(n/1) f(182) ORIGINAL a times divisor = 2184

Mobius inversion ORIGINAL inverse F(n/1) f(182) plus ORIGINAL b := sum(2184, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(182) plus ORIGINAL b = 2185

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(13, 25, 85, 157, 169, 313, 1093, 2185)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 4040

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 4040)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 4040

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(182, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 91

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=91;check complete integer range1..91

ALL positive divisors INCREASING=[1, 7, 13, 91];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(12, 1)

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

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(12, 7)

Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 84

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(84, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 85

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor := mul(12, 13)

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 156

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b := sum(156, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 157

Mobius inversion ORIGINAL inverse F(n/2) f(91) ORIGINAL a times divisor := mul(12, 91)

Mobius inversion ORIGINAL inverse F(n/2) f(91) ORIGINAL a times divisor = 1092

Mobius inversion ORIGINAL inverse F(n/2) f(91) plus ORIGINAL b := sum(1092, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(91) plus ORIGINAL b = 1093

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(13, 85, 157, 1093)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 1348

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 1348)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -1348

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7

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

ORIGINAL positive integer absolute magnitude=7>1

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

next divisor=2;previous remaining=7;all smaller divisors exhausted

extracted multiplicity=0;next remaining=7;remaining modulo2=1;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 7)

upf canonical complete product factor0 = 7

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7

complete increasing prime multiset=[7];full product=7;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(182, 7)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 26

Mobius inversion ORIGINAL inverse F(n/7):actual positive input=26;check complete integer range1..26

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

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(12, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 13

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(12, 2)

Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/7) f(13) ORIGINAL a times divisor := mul(12, 13)

Mobius inversion ORIGINAL inverse F(n/7) f(13) ORIGINAL a times divisor = 156

Mobius inversion ORIGINAL inverse F(n/7) f(13) plus ORIGINAL b := sum(156, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(13) plus ORIGINAL b = 157

Mobius inversion ORIGINAL inverse F(n/7) f(26) ORIGINAL a times divisor := mul(12, 26)

Mobius inversion ORIGINAL inverse F(n/7) f(26) ORIGINAL a times divisor = 312

Mobius inversion ORIGINAL inverse F(n/7) f(26) plus ORIGINAL b := sum(312, 1)

Mobius inversion ORIGINAL inverse F(n/7) f(26) plus ORIGINAL b = 313

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(13, 25, 157, 313)

Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 508

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 508)

Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -508

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

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

ORIGINAL positive integer absolute magnitude=13>1

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

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

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

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

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

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

upf canonical complete product factor0 = 13

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 := div(182, 13)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 14

Mobius inversion ORIGINAL inverse F(n/13):actual positive input=14;check complete integer range1..14

ALL positive divisors INCREASING=[1, 2, 7, 14];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(12, 1)

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

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(12, 2)

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

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/13) f(7) ORIGINAL a times divisor := mul(12, 7)

Mobius inversion ORIGINAL inverse F(n/13) f(7) ORIGINAL a times divisor = 84

Mobius inversion ORIGINAL inverse F(n/13) f(7) plus ORIGINAL b := sum(84, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(7) plus ORIGINAL b = 85

Mobius inversion ORIGINAL inverse F(n/13) f(14) ORIGINAL a times divisor := mul(12, 14)

Mobius inversion ORIGINAL inverse F(n/13) f(14) ORIGINAL a times divisor = 168

Mobius inversion ORIGINAL inverse F(n/13) f(14) plus ORIGINAL b := sum(168, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(14) plus ORIGINAL b = 169

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(13, 25, 85, 169)

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 292

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 292)

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -292

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14

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

ORIGINAL positive integer absolute magnitude=14>1

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

upf extract divisor2 occurrence0 := div(14, 2)

upf extract divisor2 occurrence0 = 7

next divisor=2;previous remaining=14;all smaller divisors exhausted

extracted multiplicity=1;next remaining=7;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, 7)

upf canonical complete product factor1 = 14

exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14

complete increasing prime multiset=[2, 7];full product=14;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(182, 14)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 13

Mobius inversion ORIGINAL inverse F(n/14):actual positive input=13;check complete integer range1..13

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

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(12, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 13

Mobius inversion ORIGINAL inverse F(n/14) f(13) ORIGINAL a times divisor := mul(12, 13)

Mobius inversion ORIGINAL inverse F(n/14) f(13) ORIGINAL a times divisor = 156

Mobius inversion ORIGINAL inverse F(n/14) f(13) plus ORIGINAL b := sum(156, 1)

Mobius inversion ORIGINAL inverse F(n/14) f(13) plus ORIGINAL b = 157

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(13, 157)

Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 170

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 170)

Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 170

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

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

ORIGINAL positive integer absolute magnitude=26>1

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

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

upf extract divisor2 occurrence0 = 13

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

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

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

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 26

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 := div(182, 26)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 7

Mobius inversion ORIGINAL inverse F(n/26):actual positive input=7;check complete integer range1..7

ALL positive divisors INCREASING=[1, 7];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(12, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b = 13

Mobius inversion ORIGINAL inverse F(n/26) f(7) ORIGINAL a times divisor := mul(12, 7)

Mobius inversion ORIGINAL inverse F(n/26) f(7) ORIGINAL a times divisor = 84

Mobius inversion ORIGINAL inverse F(n/26) f(7) plus ORIGINAL b := sum(84, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(7) plus ORIGINAL b = 85

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum := sum(13, 85)

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 98

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) := mul(1, 98)

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 98

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor91:actual positive input for mu=91

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

ORIGINAL positive integer absolute magnitude=91>1

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

next divisor=2;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo4=3;extracted multiset=[]

next divisor=5;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo5=1;extracted multiset=[]

next divisor=6;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo6=1;extracted multiset=[]

upf extract divisor7 occurrence0 := div(91, 7)

upf extract divisor7 occurrence0 = 13

next divisor=7;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo7=6;extracted multiset=[7]

upf canonical complete product factor0 := mul(1, 7)

upf canonical complete product factor0 = 7

upf canonical complete product factor1 := mul(7, 13)

upf canonical complete product factor1 = 91

exhausted trial divisors below8;terminal remaining=13;d*d>remaining;ORIGINAL n=91

complete increasing prime multiset=[7, 13];full product=91;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor91:input=91;COMPLETE prime multiset=[7, 13];distinct primes=[7, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor91 := div(182, 91)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor91 = 2

Mobius inversion ORIGINAL inverse F(n/91):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/91) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/91) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/91) f(1) plus ORIGINAL b := sum(12, 1)

Mobius inversion ORIGINAL inverse F(n/91) f(1) plus ORIGINAL b = 13

Mobius inversion ORIGINAL inverse F(n/91) f(2) ORIGINAL a times divisor := mul(12, 2)

Mobius inversion ORIGINAL inverse F(n/91) f(2) ORIGINAL a times divisor = 24

Mobius inversion ORIGINAL inverse F(n/91) f(2) plus ORIGINAL b := sum(24, 1)

Mobius inversion ORIGINAL inverse F(n/91) f(2) plus ORIGINAL b = 25

Mobius inversion ORIGINAL inverse F(n/91) full forward divisor sum := sum(13, 25)

Mobius inversion ORIGINAL inverse F(n/91) full forward divisor sum = 38

Mobius inversion ORIGINAL inverse mu(91) times SAME F(n/91) := mul(1, 38)

Mobius inversion ORIGINAL inverse mu(91) times SAME F(n/91) = 38

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor182:actual positive input for mu=182

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

ORIGINAL positive integer absolute magnitude=182>1

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

upf extract divisor2 occurrence0 := div(182, 2)

upf extract divisor2 occurrence0 = 91

next divisor=2;previous remaining=182;all smaller divisors exhausted

extracted multiplicity=1;next remaining=91;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo4=3;extracted multiset=[2]

next divisor=5;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo5=1;extracted multiset=[2]

next divisor=6;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=0;next remaining=91;remaining modulo6=1;extracted multiset=[2]

upf extract divisor7 occurrence1 := div(91, 7)

upf extract divisor7 occurrence1 = 13

next divisor=7;previous remaining=91;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo7=6;extracted multiset=[2, 7]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 7)

upf canonical complete product factor1 = 14

upf canonical complete product factor2 := mul(14, 13)

upf canonical complete product factor2 = 182

exhausted trial divisors below8;terminal remaining=13;d*d>remaining;ORIGINAL n=182

complete increasing prime multiset=[2, 7, 13];full product=182;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor182:input=182;COMPLETE prime multiset=[2, 7, 13];distinct primes=[2, 7, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor182 := div(182, 182)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor182 = 1

Mobius inversion ORIGINAL inverse F(n/182):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/182) f(1) ORIGINAL a times divisor := mul(12, 1)

Mobius inversion ORIGINAL inverse F(n/182) f(1) ORIGINAL a times divisor = 12

Mobius inversion ORIGINAL inverse F(n/182) f(1) plus ORIGINAL b := sum(12, 1)

Mobius inversion ORIGINAL inverse F(n/182) f(1) plus ORIGINAL b = 13

Mobius inversion ORIGINAL inverse F(n/182) full forward divisor sum := sum(13)

Mobius inversion ORIGINAL inverse F(n/182) full forward divisor sum = 13

Mobius inversion ORIGINAL inverse mu(182) times SAME F(n/182) := mul(-1, 13)

Mobius inversion ORIGINAL inverse mu(182) times SAME F(n/182) = -13

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(4040, -1348, -508, -292, 170, 98, 38, -13)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 2185

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(12, 182)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 2184

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(2184, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 2185

Mobius inversion ORIGINAL inverse:actual n=182;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 182, 4040, 4040], [2, -1, 91, 1348, -1348], [7, -1, 26, 508, -508], [13, -1, 14, 292, -292], [14, 1, 13, 170, 170], [26, 1, 7, 98, 98], [91, 1, 2, 38, 38], [182, -1, 1, 13, -13]]

complete reconstructed=2185;direct f(n)=2185;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(-1348, -508, -292, 170, 98, 38, -13)

Mobius inversion ORIGINAL omitted-divisor partial sum = -1855

Mobius inversion complete minus partial := sub(2185, -1855)

Mobius inversion complete minus partial = 4040

ORIGINAL full divisors=[1, 2, 7, 13, 14, 26, 91, 182];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 7, 13, 14, 26, 91, 182]

complete=2185=f(n)=2185;partial=-1855;lost term=4040!=0;incomplete_certificate fixed No

逐步反向核對

• S160 · 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。
• S159 · 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。
• S158 · 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。
• S157 · 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。
• S156 · 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。
• S155 · 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。
• S154 · 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。
• S153 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(182) times SAME F(n/182)，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/182) full forward divisor sum，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 F(n/182) f(1) plus ORIGINAL b，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 F(n/182) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S149 · 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。
• S148 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor182，不以小數近似替代。
• S147 · 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。
• S146 · 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。
• S145 · 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。
• S144 · 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。
• S143 · 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。
• 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 extract divisor7 occurrence1，不以小數近似替代。
• S140 · 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。
• S139 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S132 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(91) times SAME F(n/91)，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/91) full forward divisor sum，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/91) f(2) plus ORIGINAL b，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/91) f(2) ORIGINAL a times divisor，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/91) f(1) 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/91) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S126 · 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。
• S125 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor91，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S122 · 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。
• S121 · 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。
• S120 · 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。
• S119 · premise_and_result_audit — 以精確有理數重算 upf extract divisor7 occurrence0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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(26) times SAME F(n/26)，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/26) 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/26) f(7) 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/26) f(7) 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/26) 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/26) 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 divisor26，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(14) times SAME F(n/14)，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/14) 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/14) f(13) 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/14) f(13) 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/14) f(1) 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/14) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S088 · 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。
• S087 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14，不以小數近似替代。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• 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 — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S080 · 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。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13)，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(14) plus ORIGINAL b，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(14) ORIGINAL a times divisor，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/13) f(7) plus ORIGINAL b，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/13) f(7) ORIGINAL a times divisor，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/13) f(2) plus ORIGINAL b，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/13) f(2) ORIGINAL a times divisor，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/13) f(1) plus ORIGINAL b，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S068 · 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。
• S067 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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(7) times SAME F(n/7)，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/7) 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/7) f(26) 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/7) f(26) 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/7) f(13) 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/7) f(13) 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/7) 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/7) 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/7) 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/7) 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 divisor7，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(91) 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(91) 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(13) 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(13) 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(7) 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(7) 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(182) 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(182) 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(91) 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(91) 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(26) 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(26) 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(14) 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(14) 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(13) 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(13) 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(7) 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(7) 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

(182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, 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。