第 22146 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":506,"a":21,"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":506,"a":21,"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=506;check complete integer range1..506

ALL positive divisors INCREASING=[1, 2, 11, 22, 23, 46, 253, 506];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(506, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 506

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=506;check complete integer range1..506

ALL positive divisors INCREASING=[1, 2, 11, 22, 23, 46, 253, 506];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(21, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(21, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 42

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(42, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 43

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(21, 11)

Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 231

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(231, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 232

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(21, 22)

Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 462

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(462, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 463

Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor := mul(21, 23)

Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor = 483

Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b := sum(483, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b = 484

Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor := mul(21, 46)

Mobius inversion ORIGINAL inverse F(n/1) f(46) ORIGINAL a times divisor = 966

Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b := sum(966, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(46) plus ORIGINAL b = 967

Mobius inversion ORIGINAL inverse F(n/1) f(253) ORIGINAL a times divisor := mul(21, 253)

Mobius inversion ORIGINAL inverse F(n/1) f(253) ORIGINAL a times divisor = 5313

Mobius inversion ORIGINAL inverse F(n/1) f(253) plus ORIGINAL b := sum(5313, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(253) plus ORIGINAL b = 5314

Mobius inversion ORIGINAL inverse F(n/1) f(506) ORIGINAL a times divisor := mul(21, 506)

Mobius inversion ORIGINAL inverse F(n/1) f(506) ORIGINAL a times divisor = 10626

Mobius inversion ORIGINAL inverse F(n/1) f(506) plus ORIGINAL b := sum(10626, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(506) plus ORIGINAL b = 10627

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(22, 43, 232, 463, 484, 967, 5314, 10627)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 18152

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 18152)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 18152

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(506, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 253

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=253;check complete integer range1..253

ALL positive divisors INCREASING=[1, 11, 23, 253];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(21, 11)

Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 231

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(231, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 232

Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor := mul(21, 23)

Mobius inversion ORIGINAL inverse F(n/2) f(23) ORIGINAL a times divisor = 483

Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b := sum(483, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(23) plus ORIGINAL b = 484

Mobius inversion ORIGINAL inverse F(n/2) f(253) ORIGINAL a times divisor := mul(21, 253)

Mobius inversion ORIGINAL inverse F(n/2) f(253) ORIGINAL a times divisor = 5313

Mobius inversion ORIGINAL inverse F(n/2) f(253) plus ORIGINAL b := sum(5313, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(253) plus ORIGINAL b = 5314

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(22, 232, 484, 5314)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 6052

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 6052)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -6052

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(506, 11)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 46

Mobius inversion ORIGINAL inverse F(n/11):actual positive input=46;check complete integer range1..46

ALL positive divisors INCREASING=[1, 2, 23, 46];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(21, 2)

Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 42

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(42, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 43

Mobius inversion ORIGINAL inverse F(n/11) f(23) ORIGINAL a times divisor := mul(21, 23)

Mobius inversion ORIGINAL inverse F(n/11) f(23) ORIGINAL a times divisor = 483

Mobius inversion ORIGINAL inverse F(n/11) f(23) plus ORIGINAL b := sum(483, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(23) plus ORIGINAL b = 484

Mobius inversion ORIGINAL inverse F(n/11) f(46) ORIGINAL a times divisor := mul(21, 46)

Mobius inversion ORIGINAL inverse F(n/11) f(46) ORIGINAL a times divisor = 966

Mobius inversion ORIGINAL inverse F(n/11) f(46) plus ORIGINAL b := sum(966, 1)

Mobius inversion ORIGINAL inverse F(n/11) f(46) plus ORIGINAL b = 967

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(22, 43, 484, 967)

Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 1516

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 1516)

Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -1516

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(506, 22)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 23

Mobius inversion ORIGINAL inverse F(n/22):actual positive input=23;check complete integer range1..23

ALL positive divisors INCREASING=[1, 23];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(21, 1)

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

Mobius inversion ORIGINAL inverse F(n/22) f(23) ORIGINAL a times divisor := mul(21, 23)

Mobius inversion ORIGINAL inverse F(n/22) f(23) ORIGINAL a times divisor = 483

Mobius inversion ORIGINAL inverse F(n/22) f(23) plus ORIGINAL b := sum(483, 1)

Mobius inversion ORIGINAL inverse F(n/22) f(23) plus ORIGINAL b = 484

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(22, 484)

Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 506

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 506)

Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 506

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:actual positive input for mu=23

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

ORIGINAL positive integer absolute magnitude=23>1

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

next divisor=2;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[]

upf canonical complete product factor0 := mul(1, 23)

upf canonical complete product factor0 = 23

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=23

complete increasing prime multiset=[23];full product=23;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:input=23;COMPLETE prime multiset=[23];distinct primes=[23]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 := div(506, 23)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 = 22

Mobius inversion ORIGINAL inverse F(n/23):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/23) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor := mul(21, 2)

Mobius inversion ORIGINAL inverse F(n/23) f(2) ORIGINAL a times divisor = 42

Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b := sum(42, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(2) plus ORIGINAL b = 43

Mobius inversion ORIGINAL inverse F(n/23) f(11) ORIGINAL a times divisor := mul(21, 11)

Mobius inversion ORIGINAL inverse F(n/23) f(11) ORIGINAL a times divisor = 231

Mobius inversion ORIGINAL inverse F(n/23) f(11) plus ORIGINAL b := sum(231, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(11) plus ORIGINAL b = 232

Mobius inversion ORIGINAL inverse F(n/23) f(22) ORIGINAL a times divisor := mul(21, 22)

Mobius inversion ORIGINAL inverse F(n/23) f(22) ORIGINAL a times divisor = 462

Mobius inversion ORIGINAL inverse F(n/23) f(22) plus ORIGINAL b := sum(462, 1)

Mobius inversion ORIGINAL inverse F(n/23) f(22) plus ORIGINAL b = 463

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum := sum(22, 43, 232, 463)

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum = 760

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) := mul(-1, 760)

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) = -760

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:actual positive input for mu=46

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

ORIGINAL positive integer absolute magnitude=46>1

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

upf extract divisor2 occurrence0 := div(46, 2)

upf extract divisor2 occurrence0 = 23

next divisor=2;previous remaining=46;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=23;all smaller divisors exhausted

extracted multiplicity=0;next remaining=23;remaining modulo4=3;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 23)

upf canonical complete product factor1 = 46

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=46

complete increasing prime multiset=[2, 23];full product=46;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor46:input=46;COMPLETE prime multiset=[2, 23];distinct primes=[2, 23]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 := div(506, 46)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46 = 11

Mobius inversion ORIGINAL inverse F(n/46):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/46) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/46) f(11) ORIGINAL a times divisor := mul(21, 11)

Mobius inversion ORIGINAL inverse F(n/46) f(11) ORIGINAL a times divisor = 231

Mobius inversion ORIGINAL inverse F(n/46) f(11) plus ORIGINAL b := sum(231, 1)

Mobius inversion ORIGINAL inverse F(n/46) f(11) plus ORIGINAL b = 232

Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum := sum(22, 232)

Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum = 254

Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) := mul(1, 254)

Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46) = 254

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor253:actual positive input for mu=253

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

ORIGINAL positive integer absolute magnitude=253>1

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

next divisor=2;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo2=1;extracted multiset=[]

next divisor=3;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo3=1;extracted multiset=[]

next divisor=4;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo4=1;extracted multiset=[]

next divisor=5;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo5=3;extracted multiset=[]

next divisor=6;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo6=1;extracted multiset=[]

next divisor=7;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo7=1;extracted multiset=[]

next divisor=8;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo8=5;extracted multiset=[]

next divisor=9;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo9=1;extracted multiset=[]

next divisor=10;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo10=3;extracted multiset=[]

upf extract divisor11 occurrence0 := div(253, 11)

upf extract divisor11 occurrence0 = 23

next divisor=11;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo11=1;extracted multiset=[11]

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

upf canonical complete product factor0 = 11

upf canonical complete product factor1 := mul(11, 23)

upf canonical complete product factor1 = 253

exhausted trial divisors below12;terminal remaining=23;d*d>remaining;ORIGINAL n=253

complete increasing prime multiset=[11, 23];full product=253;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor253:input=253;COMPLETE prime multiset=[11, 23];distinct primes=[11, 23]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor253 := div(506, 253)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor253 = 2

Mobius inversion ORIGINAL inverse F(n/253):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/253) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/253) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/253) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/253) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/253) f(2) ORIGINAL a times divisor := mul(21, 2)

Mobius inversion ORIGINAL inverse F(n/253) f(2) ORIGINAL a times divisor = 42

Mobius inversion ORIGINAL inverse F(n/253) f(2) plus ORIGINAL b := sum(42, 1)

Mobius inversion ORIGINAL inverse F(n/253) f(2) plus ORIGINAL b = 43

Mobius inversion ORIGINAL inverse F(n/253) full forward divisor sum := sum(22, 43)

Mobius inversion ORIGINAL inverse F(n/253) full forward divisor sum = 65

Mobius inversion ORIGINAL inverse mu(253) times SAME F(n/253) := mul(1, 65)

Mobius inversion ORIGINAL inverse mu(253) times SAME F(n/253) = 65

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor506:actual positive input for mu=506

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

ORIGINAL positive integer absolute magnitude=506>1

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

upf extract divisor2 occurrence0 := div(506, 2)

upf extract divisor2 occurrence0 = 253

next divisor=2;previous remaining=506;all smaller divisors exhausted

extracted multiplicity=1;next remaining=253;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo3=1;extracted multiset=[2]

next divisor=4;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo4=1;extracted multiset=[2]

next divisor=5;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo5=3;extracted multiset=[2]

next divisor=6;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo6=1;extracted multiset=[2]

next divisor=7;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo7=1;extracted multiset=[2]

next divisor=8;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo8=5;extracted multiset=[2]

next divisor=9;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo9=1;extracted multiset=[2]

next divisor=10;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=0;next remaining=253;remaining modulo10=3;extracted multiset=[2]

upf extract divisor11 occurrence1 := div(253, 11)

upf extract divisor11 occurrence1 = 23

next divisor=11;previous remaining=253;all smaller divisors exhausted

extracted multiplicity=1;next remaining=23;remaining modulo11=1;extracted multiset=[2, 11]

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

upf canonical complete product factor2 := mul(22, 23)

upf canonical complete product factor2 = 506

exhausted trial divisors below12;terminal remaining=23;d*d>remaining;ORIGINAL n=506

complete increasing prime multiset=[2, 11, 23];full product=506;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor506:input=506;COMPLETE prime multiset=[2, 11, 23];distinct primes=[2, 11, 23]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor506 := div(506, 506)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor506 = 1

Mobius inversion ORIGINAL inverse F(n/506):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/506) f(1) ORIGINAL a times divisor := mul(21, 1)

Mobius inversion ORIGINAL inverse F(n/506) f(1) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL inverse F(n/506) f(1) plus ORIGINAL b := sum(21, 1)

Mobius inversion ORIGINAL inverse F(n/506) f(1) plus ORIGINAL b = 22

Mobius inversion ORIGINAL inverse F(n/506) full forward divisor sum := sum(22)

Mobius inversion ORIGINAL inverse F(n/506) full forward divisor sum = 22

Mobius inversion ORIGINAL inverse mu(506) times SAME F(n/506) := mul(-1, 22)

Mobius inversion ORIGINAL inverse mu(506) times SAME F(n/506) = -22

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(18152, -6052, -1516, 506, -760, 254, 65, -22)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 10627

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(21, 506)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 10626

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(10626, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 10627

Mobius inversion ORIGINAL inverse:actual n=506;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 506, 18152, 18152], [2, -1, 253, 6052, -6052], [11, -1, 46, 1516, -1516], [22, 1, 23, 506, 506], [23, -1, 22, 760, -760], [46, 1, 11, 254, 254], [253, 1, 2, 65, 65], [506, -1, 1, 22, -22]]

complete reconstructed=10627;direct f(n)=10627;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(-6052, -1516, 506, -760, 254, 65, -22)

Mobius inversion ORIGINAL omitted-divisor partial sum = -7525

Mobius inversion complete minus partial := sub(10627, -7525)

Mobius inversion complete minus partial = 18152

ORIGINAL full divisors=[1, 2, 11, 22, 23, 46, 253, 506];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 11, 22, 23, 46, 253, 506]

complete=10627=f(n)=10627;partial=-7525;lost term=18152!=0;incomplete_certificate fixed No

逐步反向核對

• S172 · 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。
• S171 · 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。
• S170 · 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。
• S169 · 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。
• S168 · 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。
• S167 · 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。
• S166 · 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。
• S165 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(506) times SAME F(n/506)，不以小數近似替代。
• S164 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/506) full forward divisor sum，不以小數近似替代。
• S163 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/506) f(1) plus ORIGINAL b，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/506) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S161 · 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。
• S160 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor506，不以小數近似替代。
• S159 · 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。
• S158 · 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。
• S157 · 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。
• S156 · 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。
• S155 · 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。
• S154 · 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。
• S153 · premise_and_result_audit — 以精確有理數重算 upf extract divisor11 occurrence1，不以小數近似替代。
• S152 · 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。
• S151 · 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。
• S150 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S143 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S140 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(253) times SAME F(n/253)，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/253) full forward divisor sum，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/253) f(2) plus ORIGINAL b，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/253) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/253) f(1) plus ORIGINAL b，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/253) f(1) ORIGINAL a times divisor，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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor253，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 upf canonical complete product factor1，不以小數近似替代。
• S129 · 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。
• S128 · 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。
• S127 · premise_and_result_audit — 以精確有理數重算 upf extract divisor11 occurrence0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(46) times SAME F(n/46)，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) full forward divisor sum，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(11) plus ORIGINAL b，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(11) ORIGINAL a times divisor，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/46) f(1) plus ORIGINAL b，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/46) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S109 · 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。
• S108 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor46，不以小數近似替代。
• S107 · 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。
• S106 · 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。
• S105 · 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。
• S104 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S100 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，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 Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23)，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(22) plus ORIGINAL b，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/23) f(22) ORIGINAL a times divisor，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/23) f(11) plus ORIGINAL b，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/23) f(11) ORIGINAL a times divisor，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/23) f(2) plus ORIGINAL b，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/23) f(2) ORIGINAL a times divisor，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/23) f(1) plus ORIGINAL b，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/23) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S087 · 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。
• S086 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23，不以小數近似替代。
• 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 — 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 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 — 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 — 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 — 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 — 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。
• S076 · 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。
• S075 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(23) plus ORIGINAL b，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/22) f(23) ORIGINAL a times divisor，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/22) f(1) 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/22) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S071 · 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。
• S070 · 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。
• 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 — 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 upf canonical complete product factor1，不以小數近似替代。
• S066 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• 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 — 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。
• S059 · 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。
• S058 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(46) plus ORIGINAL b，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/11) f(46) ORIGINAL a times divisor，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/11) f(23) plus ORIGINAL b，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/11) f(23) ORIGINAL a times divisor，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/11) f(2) plus ORIGINAL b，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/11) f(2) ORIGINAL a times divisor，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/11) f(1) plus ORIGINAL b，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/11) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• 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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S045 · 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。
• 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(253) 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(253) 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(23) 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(23) 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(11) 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(11) 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(506) 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(506) 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(253) 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(253) 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(46) 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(46) 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(23) 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(23) 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(22) 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(22) 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(11) 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(11) 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

(506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, 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。