第 22150 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":702,"a":25,"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":702,"a":25,"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=702;check complete integer range1..702

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];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(702, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 702

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=702;check complete integer range1..702

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(25, 1)

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/1) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor := mul(25, 18)

Mobius inversion ORIGINAL inverse F(n/1) f(18) ORIGINAL a times divisor = 450

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b := sum(450, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(18) plus ORIGINAL b = 451

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor := mul(25, 26)

Mobius inversion ORIGINAL inverse F(n/1) f(26) ORIGINAL a times divisor = 650

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b := sum(650, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(26) plus ORIGINAL b = 651

Mobius inversion ORIGINAL inverse F(n/1) f(27) ORIGINAL a times divisor := mul(25, 27)

Mobius inversion ORIGINAL inverse F(n/1) f(27) ORIGINAL a times divisor = 675

Mobius inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b := sum(675, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b = 676

Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor := mul(25, 54)

Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor = 1350

Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b := sum(1350, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b = 1351

Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor := mul(25, 78)

Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor = 1950

Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b := sum(1950, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b = 1951

Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor := mul(25, 117)

Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor = 2925

Mobius inversion ORIGINAL inverse F(n/1) f(117) plus ORIGINAL b := sum(2925, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(117) plus ORIGINAL b = 2926

Mobius inversion ORIGINAL inverse F(n/1) f(234) ORIGINAL a times divisor := mul(25, 234)

Mobius inversion ORIGINAL inverse F(n/1) f(234) ORIGINAL a times divisor = 5850

Mobius inversion ORIGINAL inverse F(n/1) f(234) plus ORIGINAL b := sum(5850, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(234) plus ORIGINAL b = 5851

Mobius inversion ORIGINAL inverse F(n/1) f(351) ORIGINAL a times divisor := mul(25, 351)

Mobius inversion ORIGINAL inverse F(n/1) f(351) ORIGINAL a times divisor = 8775

Mobius inversion ORIGINAL inverse F(n/1) f(351) plus ORIGINAL b := sum(8775, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(351) plus ORIGINAL b = 8776

Mobius inversion ORIGINAL inverse F(n/1) f(702) ORIGINAL a times divisor := mul(25, 702)

Mobius inversion ORIGINAL inverse F(n/1) f(702) ORIGINAL a times divisor = 17550

Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b := sum(17550, 1)

Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b = 17551

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(26, 51, 76, 151, 226, 326, 451, 651, 676, 976, 1351, 1951, 2926, 5851, 8776, 17551)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 42016

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 42016)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 42016

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(702, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 351

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=351;check complete integer range1..351

ALL positive divisors INCREASING=[1, 3, 9, 13, 27, 39, 117, 351];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 25

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

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/2) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/2) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/2) f(27) ORIGINAL a times divisor := mul(25, 27)

Mobius inversion ORIGINAL inverse F(n/2) f(27) ORIGINAL a times divisor = 675

Mobius inversion ORIGINAL inverse F(n/2) f(27) plus ORIGINAL b := sum(675, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(27) plus ORIGINAL b = 676

Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/2) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor := mul(25, 117)

Mobius inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor = 2925

Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b := sum(2925, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b = 2926

Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor := mul(25, 351)

Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor = 8775

Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b := sum(8775, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b = 8776

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(26, 76, 226, 326, 676, 976, 2926, 8776)

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 14008

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 14008)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -14008

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3

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

ORIGINAL positive integer absolute magnitude=3>1

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

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3

complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(702, 3)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 234

Mobius inversion ORIGINAL inverse F(n/3):actual positive input=234;check complete integer range1..234

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/3) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/3) f(18) ORIGINAL a times divisor := mul(25, 18)

Mobius inversion ORIGINAL inverse F(n/3) f(18) ORIGINAL a times divisor = 450

Mobius inversion ORIGINAL inverse F(n/3) f(18) plus ORIGINAL b := sum(450, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(18) plus ORIGINAL b = 451

Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor := mul(25, 26)

Mobius inversion ORIGINAL inverse F(n/3) f(26) ORIGINAL a times divisor = 650

Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b := sum(650, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(26) plus ORIGINAL b = 651

Mobius inversion ORIGINAL inverse F(n/3) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/3) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/3) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/3) f(78) ORIGINAL a times divisor := mul(25, 78)

Mobius inversion ORIGINAL inverse F(n/3) f(78) ORIGINAL a times divisor = 1950

Mobius inversion ORIGINAL inverse F(n/3) f(78) plus ORIGINAL b := sum(1950, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(78) plus ORIGINAL b = 1951

Mobius inversion ORIGINAL inverse F(n/3) f(117) ORIGINAL a times divisor := mul(25, 117)

Mobius inversion ORIGINAL inverse F(n/3) f(117) ORIGINAL a times divisor = 2925

Mobius inversion ORIGINAL inverse F(n/3) f(117) plus ORIGINAL b := sum(2925, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(117) plus ORIGINAL b = 2926

Mobius inversion ORIGINAL inverse F(n/3) f(234) ORIGINAL a times divisor := mul(25, 234)

Mobius inversion ORIGINAL inverse F(n/3) f(234) ORIGINAL a times divisor = 5850

Mobius inversion ORIGINAL inverse F(n/3) f(234) plus ORIGINAL b := sum(5850, 1)

Mobius inversion ORIGINAL inverse F(n/3) f(234) plus ORIGINAL b = 5851

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(26, 51, 76, 151, 226, 326, 451, 651, 976, 1951, 2926, 5851)

Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 13662

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 13662)

Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -13662

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6

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

ORIGINAL positive integer absolute magnitude=6>1

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

upf extract divisor2 occurrence0 := div(6, 2)

upf extract divisor2 occurrence0 = 3

next divisor=2;previous remaining=6;all smaller divisors exhausted

extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6

complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(702, 6)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 117

Mobius inversion ORIGINAL inverse F(n/6):actual positive input=117;check complete integer range1..117

ALL positive divisors INCREASING=[1, 3, 9, 13, 39, 117];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/6) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/6) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/6) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/6) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor := mul(25, 117)

Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor = 2925

Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b := sum(2925, 1)

Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b = 2926

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(26, 76, 226, 326, 976, 2926)

Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 4556

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 4556)

Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 4556

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:actual positive input for mu=9

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

ORIGINAL positive integer absolute magnitude=9>1

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

next divisor=2;previous remaining=9;all smaller divisors exhausted

extracted multiplicity=0;next remaining=9;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(9, 3)

upf extract divisor3 occurrence0 = 3

upf extract divisor3 occurrence1 := div(3, 3)

upf extract divisor3 occurrence1 = 1

next divisor=3;previous remaining=9;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo3=1;extracted multiset=[3, 3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 3)

upf canonical complete product factor1 = 9

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=9

complete increasing prime multiset=[3, 3];full product=9;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor9:input=9;COMPLETE prime multiset=[3, 3];distinct primes=[3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 := div(702, 9)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor9 = 78

Mobius inversion ORIGINAL inverse F(n/9):actual positive input=78;check complete integer range1..78

ALL positive divisors INCREASING=[1, 2, 3, 6, 13, 26, 39, 78];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/9) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/9) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/9) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor := mul(25, 26)

Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor = 650

Mobius inversion ORIGINAL inverse F(n/9) f(26) plus ORIGINAL b := sum(650, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(26) plus ORIGINAL b = 651

Mobius inversion ORIGINAL inverse F(n/9) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/9) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/9) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/9) f(78) ORIGINAL a times divisor := mul(25, 78)

Mobius inversion ORIGINAL inverse F(n/9) f(78) ORIGINAL a times divisor = 1950

Mobius inversion ORIGINAL inverse F(n/9) f(78) plus ORIGINAL b := sum(1950, 1)

Mobius inversion ORIGINAL inverse F(n/9) f(78) plus ORIGINAL b = 1951

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum := sum(26, 51, 76, 151, 326, 651, 976, 1951)

Mobius inversion ORIGINAL inverse F(n/9) full forward divisor sum = 4208

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) := mul(0, 4208)

Mobius inversion ORIGINAL inverse mu(9) times SAME F(n/9) = 0

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(702, 13)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 54

Mobius inversion ORIGINAL inverse F(n/13):actual positive input=54;check complete integer range1..54

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18, 27, 54];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(25, 1)

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

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/13) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/13) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor := mul(25, 18)

Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor = 450

Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b := sum(450, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b = 451

Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor := mul(25, 27)

Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor = 675

Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b := sum(675, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b = 676

Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor := mul(25, 54)

Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor = 1350

Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b := sum(1350, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b = 1351

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(26, 51, 76, 151, 226, 451, 676, 1351)

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 3008

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 3008)

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -3008

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:actual positive input for mu=18

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

ORIGINAL positive integer absolute magnitude=18>1

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

upf extract divisor2 occurrence0 := div(18, 2)

upf extract divisor2 occurrence0 = 9

next divisor=2;previous remaining=18;all smaller divisors exhausted

extracted multiplicity=1;next remaining=9;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(9, 3)

upf extract divisor3 occurrence1 = 3

upf extract divisor3 occurrence2 := div(3, 3)

upf extract divisor3 occurrence2 = 1

next divisor=3;previous remaining=9;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo3=1;extracted multiset=[2, 3, 3]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=18

complete increasing prime multiset=[2, 3, 3];full product=18;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor18:input=18;COMPLETE prime multiset=[2, 3, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 := div(702, 18)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18 = 39

Mobius inversion ORIGINAL inverse F(n/18):actual positive input=39;check complete integer range1..39

ALL positive divisors INCREASING=[1, 3, 13, 39];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor := mul(25, 39)

Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor = 975

Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b := sum(975, 1)

Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b = 976

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum := sum(26, 76, 326, 976)

Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum = 1404

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) := mul(0, 1404)

Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18) = 0

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(702, 26)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor26 = 27

Mobius inversion ORIGINAL inverse F(n/26):actual positive input=27;check complete integer range1..27

ALL positive divisors INCREASING=[1, 3, 9, 27];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/26) f(1) plus ORIGINAL b := sum(25, 1)

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

Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor := mul(25, 27)

Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor = 675

Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b := sum(675, 1)

Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b = 676

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum := sum(26, 76, 226, 676)

Mobius inversion ORIGINAL inverse F(n/26) full forward divisor sum = 1004

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) := mul(1, 1004)

Mobius inversion ORIGINAL inverse mu(26) times SAME F(n/26) = 1004

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor27:actual positive input for mu=27

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

ORIGINAL positive integer absolute magnitude=27>1

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

next divisor=2;previous remaining=27;all smaller divisors exhausted

extracted multiplicity=0;next remaining=27;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(27, 3)

upf extract divisor3 occurrence0 = 9

upf extract divisor3 occurrence1 := div(9, 3)

upf extract divisor3 occurrence1 = 3

upf extract divisor3 occurrence2 := div(3, 3)

upf extract divisor3 occurrence2 = 1

next divisor=3;previous remaining=27;all smaller divisors exhausted

extracted multiplicity=3;next remaining=1;remaining modulo3=1;extracted multiset=[3, 3, 3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 3)

upf canonical complete product factor1 = 9

upf canonical complete product factor2 := mul(9, 3)

upf canonical complete product factor2 = 27

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=27

complete increasing prime multiset=[3, 3, 3];full product=27;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor27:input=27;COMPLETE prime multiset=[3, 3, 3];distinct primes=[3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27 := div(702, 27)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27 = 26

Mobius inversion ORIGINAL inverse F(n/27):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/27) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/27) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor := mul(25, 26)

Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor = 650

Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b := sum(650, 1)

Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b = 651

Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum := sum(26, 51, 326, 651)

Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum = 1054

Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27) := mul(0, 1054)

Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:actual positive input for mu=39

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

ORIGINAL positive integer absolute magnitude=39>1

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

next divisor=2;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=0;next remaining=39;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(39, 3)

upf extract divisor3 occurrence0 = 13

next divisor=3;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo3=1;extracted multiset=[3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 13)

upf canonical complete product factor1 = 39

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=39

complete increasing prime multiset=[3, 13];full product=39;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor39:input=39;COMPLETE prime multiset=[3, 13];distinct primes=[3, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 := div(702, 39)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39 = 18

Mobius inversion ORIGINAL inverse F(n/39):actual positive input=18;check complete integer range1..18

ALL positive divisors INCREASING=[1, 2, 3, 6, 9, 18];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor := mul(25, 18)

Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor = 450

Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b := sum(450, 1)

Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b = 451

Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum := sum(26, 51, 76, 151, 226, 451)

Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum = 981

Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) := mul(1, 981)

Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39) = 981

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor54:actual positive input for mu=54

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

ORIGINAL positive integer absolute magnitude=54>1

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

upf extract divisor2 occurrence0 := div(54, 2)

upf extract divisor2 occurrence0 = 27

next divisor=2;previous remaining=54;all smaller divisors exhausted

extracted multiplicity=1;next remaining=27;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(27, 3)

upf extract divisor3 occurrence1 = 9

upf extract divisor3 occurrence2 := div(9, 3)

upf extract divisor3 occurrence2 = 3

upf extract divisor3 occurrence3 := div(3, 3)

upf extract divisor3 occurrence3 = 1

next divisor=3;previous remaining=27;all smaller divisors exhausted

extracted multiplicity=3;next remaining=1;remaining modulo3=1;extracted multiset=[2, 3, 3, 3]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

upf canonical complete product factor3 := mul(18, 3)

upf canonical complete product factor3 = 54

exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=54

complete increasing prime multiset=[2, 3, 3, 3];full product=54;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor54:input=54;COMPLETE prime multiset=[2, 3, 3, 3];distinct primes=[2, 3]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54 := div(702, 54)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54 = 13

Mobius inversion ORIGINAL inverse F(n/54):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/54) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/54) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor := mul(25, 13)

Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor = 325

Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b := sum(325, 1)

Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b = 326

Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum := sum(26, 326)

Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum = 352

Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54) := mul(0, 352)

Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:actual positive input for mu=78

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

ORIGINAL positive integer absolute magnitude=78>1

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

upf extract divisor2 occurrence0 := div(78, 2)

upf extract divisor2 occurrence0 = 39

next divisor=2;previous remaining=78;all smaller divisors exhausted

extracted multiplicity=1;next remaining=39;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(39, 3)

upf extract divisor3 occurrence1 = 13

next divisor=3;previous remaining=39;all smaller divisors exhausted

extracted multiplicity=1;next remaining=13;remaining modulo3=1;extracted multiset=[2, 3]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 13)

upf canonical complete product factor2 = 78

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=78

complete increasing prime multiset=[2, 3, 13];full product=78;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor78:input=78;COMPLETE prime multiset=[2, 3, 13];distinct primes=[2, 3, 13]

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 := div(702, 78)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78 = 9

Mobius inversion ORIGINAL inverse F(n/78):actual positive input=9;check complete integer range1..9

ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor := mul(25, 9)

Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor = 225

Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b := sum(225, 1)

Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b = 226

Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum := sum(26, 76, 226)

Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum = 328

Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) := mul(-1, 328)

Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78) = -328

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor117:actual positive input for mu=117

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

ORIGINAL positive integer absolute magnitude=117>1

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

next divisor=2;previous remaining=117;all smaller divisors exhausted

extracted multiplicity=0;next remaining=117;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(117, 3)

upf extract divisor3 occurrence0 = 39

upf extract divisor3 occurrence1 := div(39, 3)

upf extract divisor3 occurrence1 = 13

next divisor=3;previous remaining=117;all smaller divisors exhausted

extracted multiplicity=2;next remaining=13;remaining modulo3=1;extracted multiset=[3, 3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 3)

upf canonical complete product factor1 = 9

upf canonical complete product factor2 := mul(9, 13)

upf canonical complete product factor2 = 117

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=117

complete increasing prime multiset=[3, 3, 13];full product=117;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor117:input=117;COMPLETE prime multiset=[3, 3, 13];distinct primes=[3, 13]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117 := div(702, 117)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117 = 6

Mobius inversion ORIGINAL inverse F(n/117):actual positive input=6;check complete integer range1..6

ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/117) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/117) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor := mul(25, 6)

Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor = 150

Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b := sum(150, 1)

Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b = 151

Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum := sum(26, 51, 76, 151)

Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum = 304

Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117) := mul(0, 304)

Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor234:actual positive input for mu=234

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

ORIGINAL positive integer absolute magnitude=234>1

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

upf extract divisor2 occurrence0 := div(234, 2)

upf extract divisor2 occurrence0 = 117

next divisor=2;previous remaining=234;all smaller divisors exhausted

extracted multiplicity=1;next remaining=117;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(117, 3)

upf extract divisor3 occurrence1 = 39

upf extract divisor3 occurrence2 := div(39, 3)

upf extract divisor3 occurrence2 = 13

next divisor=3;previous remaining=117;all smaller divisors exhausted

extracted multiplicity=2;next remaining=13;remaining modulo3=1;extracted multiset=[2, 3, 3]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

upf canonical complete product factor3 := mul(18, 13)

upf canonical complete product factor3 = 234

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=234

complete increasing prime multiset=[2, 3, 3, 13];full product=234;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor234:input=234;COMPLETE prime multiset=[2, 3, 3, 13];distinct primes=[2, 3, 13]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234 := div(702, 234)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234 = 3

Mobius inversion ORIGINAL inverse F(n/234):actual positive input=3;check complete integer range1..3

ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL inverse F(n/234) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/234) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor := mul(25, 3)

Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor = 75

Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b := sum(75, 1)

Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b = 76

Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum := sum(26, 76)

Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum = 102

Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234) := mul(0, 102)

Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor351:actual positive input for mu=351

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

ORIGINAL positive integer absolute magnitude=351>1

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

next divisor=2;previous remaining=351;all smaller divisors exhausted

extracted multiplicity=0;next remaining=351;remaining modulo2=1;extracted multiset=[]

upf extract divisor3 occurrence0 := div(351, 3)

upf extract divisor3 occurrence0 = 117

upf extract divisor3 occurrence1 := div(117, 3)

upf extract divisor3 occurrence1 = 39

upf extract divisor3 occurrence2 := div(39, 3)

upf extract divisor3 occurrence2 = 13

next divisor=3;previous remaining=351;all smaller divisors exhausted

extracted multiplicity=3;next remaining=13;remaining modulo3=1;extracted multiset=[3, 3, 3]

upf canonical complete product factor0 := mul(1, 3)

upf canonical complete product factor0 = 3

upf canonical complete product factor1 := mul(3, 3)

upf canonical complete product factor1 = 9

upf canonical complete product factor2 := mul(9, 3)

upf canonical complete product factor2 = 27

upf canonical complete product factor3 := mul(27, 13)

upf canonical complete product factor3 = 351

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=351

complete increasing prime multiset=[3, 3, 3, 13];full product=351;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor351:input=351;COMPLETE prime multiset=[3, 3, 3, 13];distinct primes=[3, 13]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351 := div(702, 351)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351 = 2

Mobius inversion ORIGINAL inverse F(n/351):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/351) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/351) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor := mul(25, 2)

Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b := sum(50, 1)

Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b = 51

Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum := sum(26, 51)

Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum = 77

Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351) := mul(0, 77)

Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351) = 0

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor702:actual positive input for mu=702

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

ORIGINAL positive integer absolute magnitude=702>1

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

upf extract divisor2 occurrence0 := div(702, 2)

upf extract divisor2 occurrence0 = 351

next divisor=2;previous remaining=702;all smaller divisors exhausted

extracted multiplicity=1;next remaining=351;remaining modulo2=1;extracted multiset=[2]

upf extract divisor3 occurrence1 := div(351, 3)

upf extract divisor3 occurrence1 = 117

upf extract divisor3 occurrence2 := div(117, 3)

upf extract divisor3 occurrence2 = 39

upf extract divisor3 occurrence3 := div(39, 3)

upf extract divisor3 occurrence3 = 13

next divisor=3;previous remaining=351;all smaller divisors exhausted

extracted multiplicity=3;next remaining=13;remaining modulo3=1;extracted multiset=[2, 3, 3, 3]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 3)

upf canonical complete product factor1 = 6

upf canonical complete product factor2 := mul(6, 3)

upf canonical complete product factor2 = 18

upf canonical complete product factor3 := mul(18, 3)

upf canonical complete product factor3 = 54

upf canonical complete product factor4 := mul(54, 13)

upf canonical complete product factor4 = 702

exhausted trial divisors below4;terminal remaining=13;d*d>remaining;ORIGINAL n=702

complete increasing prime multiset=[2, 3, 3, 3, 13];full product=702;a composite terminal remainder would have an already tested smaller divisor

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor702:input=702;COMPLETE prime multiset=[2, 3, 3, 3, 13];distinct primes=[2, 3, 13]

mu=0;empty product case=否;repeated prime=是

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702 := div(702, 702)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702 = 1

Mobius inversion ORIGINAL inverse F(n/702):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/702) f(1) ORIGINAL a times divisor := mul(25, 1)

Mobius inversion ORIGINAL inverse F(n/702) f(1) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b := sum(25, 1)

Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b = 26

Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum := sum(26)

Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum = 26

Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702) := mul(0, 26)

Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702) = 0

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(42016, -14008, -13662, 4556, 0, -3008, 0, 1004, 0, 981, 0, -328, 0, 0, 0, 0)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 17551

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(25, 702)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 17550

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(17550, 1)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 17551

Mobius inversion ORIGINAL inverse:actual n=702;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 702, 42016, 42016], [2, -1, 351, 14008, -14008], [3, -1, 234, 13662, -13662], [6, 1, 117, 4556, 4556], [9, 0, 78, 4208, 0], [13, -1, 54, 3008, -3008], [18, 0, 39, 1404, 0], [26, 1, 27, 1004, 1004], [27, 0, 26, 1054, 0], [39, 1, 18, 981, 981], [54, 0, 13, 352, 0], [78, -1, 9, 328, -328], [117, 0, 6, 304, 0], [234, 0, 3, 102, 0], [351, 0, 2, 77, 0], [702, 0, 1, 26, 0]]

complete reconstructed=17551;direct f(n)=17551;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(-14008, -13662, 4556, 0, -3008, 0, 1004, 0, 981, 0, -328, 0, 0, 0, 0)

Mobius inversion ORIGINAL omitted-divisor partial sum = -24465

Mobius inversion complete minus partial := sub(17551, -24465)

Mobius inversion complete minus partial = 42016

ORIGINAL full divisors=[1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702]

complete=17551=f(n)=17551;partial=-24465;lost term=42016!=0;incomplete_certificate fixed No

逐步反向核對

• S409 · 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。
• S408 · 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。
• S407 · 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。
• S406 · 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。
• S405 · 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。
• S404 · 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。
• S403 · 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。
• S402 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(702) times SAME F(n/702)，不以小數近似替代。
• S401 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/702) full forward divisor sum，不以小數近似替代。
• S400 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/702) f(1) plus ORIGINAL b，不以小數近似替代。
• S399 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/702) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S398 · 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。
• S397 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor702，不以小數近似替代。
• S396 · 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。
• S395 · 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。
• S394 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor4，不以小數近似替代。
• S393 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S392 · 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。
• S391 · 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。
• S390 · 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。
• S389 · 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。
• S388 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence3，不以小數近似替代。
• S387 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S386 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S385 · 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。
• S384 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S383 · 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。
• S382 · 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。
• S381 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(351) times SAME F(n/351)，不以小數近似替代。
• S380 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/351) full forward divisor sum，不以小數近似替代。
• S379 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/351) f(2) plus ORIGINAL b，不以小數近似替代。
• S378 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/351) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S377 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/351) f(1) plus ORIGINAL b，不以小數近似替代。
• S376 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/351) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S375 · 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。
• S374 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor351，不以小數近似替代。
• S373 · 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。
• S372 · 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。
• S371 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S370 · 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。
• S369 · 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。
• S368 · 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。
• S367 · 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。
• S366 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S365 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S364 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• S363 · 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。
• S362 · 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。
• S361 · 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。
• S360 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(234) times SAME F(n/234)，不以小數近似替代。
• S359 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/234) full forward divisor sum，不以小數近似替代。
• S358 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/234) f(3) plus ORIGINAL b，不以小數近似替代。
• S357 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/234) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S356 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/234) f(1) plus ORIGINAL b，不以小數近似替代。
• S355 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/234) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S354 · 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。
• S353 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor234，不以小數近似替代。
• S352 · 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。
• S351 · 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。
• S350 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S349 · 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。
• S348 · 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。
• S347 · 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。
• S346 · 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。
• S345 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S344 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S343 · 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。
• S342 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S341 · 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。
• S340 · 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。
• S339 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(117) times SAME F(n/117)，不以小數近似替代。
• S338 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) full forward divisor sum，不以小數近似替代。
• S337 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(6) plus ORIGINAL b，不以小數近似替代。
• S336 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S335 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(3) plus ORIGINAL b，不以小數近似替代。
• S334 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S333 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(2) plus ORIGINAL b，不以小數近似替代。
• S332 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S331 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(1) plus ORIGINAL b，不以小數近似替代。
• S330 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/117) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S329 · 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。
• S328 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor117，不以小數近似替代。
• S327 · 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。
• S326 · 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。
• S325 · 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。
• S324 · 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。
• S323 · 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。
• S322 · 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。
• S321 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S320 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• S319 · 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。
• S318 · 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。
• S317 · 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。
• S316 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(78) times SAME F(n/78)，不以小數近似替代。
• S315 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) full forward divisor sum，不以小數近似替代。
• S314 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(9) plus ORIGINAL b，不以小數近似替代。
• S313 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(9) ORIGINAL a times divisor，不以小數近似替代。
• S312 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(3) plus ORIGINAL b，不以小數近似替代。
• S311 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S310 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(1) plus ORIGINAL b，不以小數近似替代。
• S309 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/78) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S308 · 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。
• S307 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor78，不以小數近似替代。
• S306 · 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。
• S305 · 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。
• S304 · 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。
• S303 · 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。
• S302 · 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。
• S301 · 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。
• S300 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S299 · 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。
• S298 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S297 · 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。
• S296 · 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。
• S295 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(54) times SAME F(n/54)，不以小數近似替代。
• S294 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/54) full forward divisor sum，不以小數近似替代。
• S293 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/54) f(13) plus ORIGINAL b，不以小數近似替代。
• S292 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/54) f(13) ORIGINAL a times divisor，不以小數近似替代。
• S291 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/54) f(1) plus ORIGINAL b，不以小數近似替代。
• S290 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/54) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S289 · 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。
• S288 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor54，不以小數近似替代。
• S287 · 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。
• S286 · 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。
• S285 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S284 · 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。
• S283 · 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。
• S282 · 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。
• S281 · 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。
• S280 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence3，不以小數近似替代。
• S279 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S278 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S277 · 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。
• S276 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S275 · 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。
• S274 · 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。
• S273 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(39) times SAME F(n/39)，不以小數近似替代。
• S272 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) full forward divisor sum，不以小數近似替代。
• S271 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(18) plus ORIGINAL b，不以小數近似替代。
• S270 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(18) ORIGINAL a times divisor，不以小數近似替代。
• S269 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(9) plus ORIGINAL b，不以小數近似替代。
• S268 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(9) ORIGINAL a times divisor，不以小數近似替代。
• S267 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(6) plus ORIGINAL b，不以小數近似替代。
• S266 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S265 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(3) plus ORIGINAL b，不以小數近似替代。
• S264 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S263 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(2) plus ORIGINAL b，不以小數近似替代。
• S262 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S261 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(1) plus ORIGINAL b，不以小數近似替代。
• S260 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/39) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S259 · 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。
• S258 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor39，不以小數近似替代。
• S257 · 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。
• S256 · 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。
• S255 · 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。
• S254 · 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。
• S253 · 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。
• S252 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• S251 · 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。
• S250 · 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。
• S249 · 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。
• S248 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(27) times SAME F(n/27)，不以小數近似替代。
• S247 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) full forward divisor sum，不以小數近似替代。
• S246 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(26) plus ORIGINAL b，不以小數近似替代。
• S245 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(26) ORIGINAL a times divisor，不以小數近似替代。
• S244 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(13) plus ORIGINAL b，不以小數近似替代。
• S243 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(13) ORIGINAL a times divisor，不以小數近似替代。
• S242 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(2) plus ORIGINAL b，不以小數近似替代。
• S241 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S240 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(1) plus ORIGINAL b，不以小數近似替代。
• S239 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/27) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S238 · 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。
• S237 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor27，不以小數近似替代。
• S236 · 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。
• S235 · 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。
• S234 · 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。
• S233 · 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。
• S232 · 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。
• S231 · 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。
• S230 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S229 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S228 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• S227 · 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。
• S226 · 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。
• S225 · 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。
• S224 · 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。
• S223 · 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。
• S222 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(27) plus ORIGINAL b，不以小數近似替代。
• S221 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(27) ORIGINAL a times divisor，不以小數近似替代。
• S220 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(9) plus ORIGINAL b，不以小數近似替代。
• S219 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(9) ORIGINAL a times divisor，不以小數近似替代。
• S218 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(3) plus ORIGINAL b，不以小數近似替代。
• S217 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/26) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S216 · 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。
• S215 · 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。
• S214 · 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。
• S213 · 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。
• S212 · 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。
• S211 · 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。
• S210 · 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。
• S209 · 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。
• S208 · 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。
• S207 · 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。
• S206 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S205 · 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。
• S204 · 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。
• S203 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(18) times SAME F(n/18)，不以小數近似替代。
• S202 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) full forward divisor sum，不以小數近似替代。
• S201 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(39) plus ORIGINAL b，不以小數近似替代。
• S200 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(39) ORIGINAL a times divisor，不以小數近似替代。
• S199 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(13) plus ORIGINAL b，不以小數近似替代。
• S198 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(13) ORIGINAL a times divisor，不以小數近似替代。
• S197 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(3) plus ORIGINAL b，不以小數近似替代。
• S196 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S195 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(1) plus ORIGINAL b，不以小數近似替代。
• S194 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/18) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S193 · 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。
• S192 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor18，不以小數近似替代。
• S191 · 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。
• S190 · 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。
• S189 · 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。
• S188 · 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。
• S187 · 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。
• S186 · 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。
• S185 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence2，不以小數近似替代。
• S184 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。
• S183 · 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。
• S182 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S181 · 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。
• S180 · 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。
• S179 · 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。
• S178 · 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。
• S177 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(54) plus ORIGINAL b，不以小數近似替代。
• S176 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(54) ORIGINAL a times divisor，不以小數近似替代。
• S175 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(27) plus ORIGINAL b，不以小數近似替代。
• S174 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(27) ORIGINAL a times divisor，不以小數近似替代。
• S173 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(18) plus ORIGINAL b，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(18) ORIGINAL a times divisor，不以小數近似替代。
• S171 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(9) plus ORIGINAL b，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(9) ORIGINAL a times divisor，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(6) plus ORIGINAL b，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/13) f(6) ORIGINAL a times divisor，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 F(n/13) f(3) plus ORIGINAL b，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 F(n/13) f(3) ORIGINAL a times divisor，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 F(n/13) f(2) plus ORIGINAL b，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/13) f(2) ORIGINAL a times divisor，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/13) 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/13) 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 divisor13，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 factor0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 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 mu(9) times SAME F(n/9)，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/9) full forward divisor sum，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/9) f(78) plus ORIGINAL b，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 F(n/9) f(78) ORIGINAL a times divisor，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/9) f(39) plus ORIGINAL b，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/9) f(39) ORIGINAL a times divisor，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/9) f(26) plus ORIGINAL b，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(26) ORIGINAL a times divisor，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(13) plus ORIGINAL b，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(13) ORIGINAL a times divisor，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(6) plus ORIGINAL b，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/9) f(3) plus ORIGINAL b，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/9) f(3) ORIGINAL a times divisor，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/9) 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/9) 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/9) 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/9) 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 divisor9，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 divisor3 occurrence1，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6)，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(117) plus ORIGINAL b，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(117) ORIGINAL a times divisor，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(39) plus ORIGINAL b，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(39) ORIGINAL a times divisor，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(13) plus ORIGINAL b，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(13) ORIGINAL a times divisor，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/6) f(9) plus ORIGINAL b，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/6) f(9) ORIGINAL a times divisor，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/6) f(3) plus ORIGINAL b，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/6) f(3) ORIGINAL a times divisor，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/6) f(1) plus ORIGINAL b，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/6) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S108 · 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。
• S107 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6，不以小數近似替代。
• 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 — 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 upf canonical complete product factor1，不以小數近似替代。
• S103 · 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。
• 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 extract divisor2 occurrence0，不以小數近似替代。
• S100 · 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。
• 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 — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum，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/3) f(234) plus ORIGINAL b，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/3) f(234) ORIGINAL a times divisor，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/3) f(117) plus ORIGINAL b，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/3) f(117) ORIGINAL a times divisor，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/3) f(78) 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/3) f(78) 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/3) f(39) 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/3) f(39) 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/3) f(26) 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/3) f(26) 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/3) f(18) 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/3) f(18) ORIGINAL a times divisor，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(13) plus ORIGINAL b，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(13) ORIGINAL a times divisor，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(9) plus ORIGINAL b，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(9) ORIGINAL a times divisor，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(6) plus ORIGINAL b，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(6) ORIGINAL a times divisor，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(3) plus ORIGINAL b，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/3) f(3) ORIGINAL a times divisor，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/3) f(2) 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/3) f(2) 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/3) f(1) 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/3) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S072 · 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。
• S071 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3，不以小數近似替代。
• S070 · 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。
• 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 upf canonical complete product factor0，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 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。
• S064 · 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。
• S063 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(351) plus ORIGINAL b，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(351) ORIGINAL a times divisor，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(117) plus ORIGINAL b，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(117) ORIGINAL a times divisor，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/2) f(39) plus ORIGINAL b，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/2) f(39) ORIGINAL a times divisor，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/2) f(27) 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/2) f(27) 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/2) 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/2) 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/2) f(9) 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/2) f(9) 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/2) f(3) 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/2) f(3) ORIGINAL a times divisor，不以小數近似替代。
• S049 · 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。
• S048 · 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。
• 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 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · 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。
• S039 · 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。
• S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(702) plus ORIGINAL b，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/1) f(702) ORIGINAL a times divisor，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/1) f(351) plus ORIGINAL b，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/1) f(351) ORIGINAL a times divisor，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/1) f(234) plus ORIGINAL b，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/1) f(234) ORIGINAL a times divisor，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/1) f(117) plus ORIGINAL b，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(117) ORIGINAL a times divisor，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(78) plus ORIGINAL b，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(78) ORIGINAL a times divisor，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(54) plus ORIGINAL b，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(54) ORIGINAL a times divisor，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(39) plus ORIGINAL b，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(39) ORIGINAL a times divisor，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(27) plus ORIGINAL b，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) f(27) ORIGINAL a times divisor，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(26) 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(26) 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(18) 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(18) 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(13) 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(13) 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(9) 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(9) 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(6) 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(6) 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(3) 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(3) 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

(702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, 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。