第 22160 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-inverse-reconstruction；資料={"n":10,"a":5,"b":-3}。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-inverse-reconstruction；資料={"n":10,"a":5,"b":-3}。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-inverse-reconstruction;original snapshot and mathematics retained

explicit ordered source output fields=[n, inverse_terms, reconstructed, f_n, equal]

Mobius inversion ORIGINAL inverse:actual positive input=10;check complete integer range1..10

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

Mobius inversion ORIGINAL inverse 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(10, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 10

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=10;check complete integer range1..10

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

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(5, 2)

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(10, -3)

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(25, -3)

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

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(5, 10)

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 50

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(50, -3)

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 47

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(2, 7, 22, 47)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 78

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

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 78

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(10, 2)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 5

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=5;check complete integer range1..5

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

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 5

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

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

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

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 25

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(25, -3)

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 22

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

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 24

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 24)

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -24

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

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

ORIGINAL positive integer absolute magnitude=5>1

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

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

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

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

upf canonical complete product factor0 = 5

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(10, 5)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 2

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

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

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

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

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

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 10

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(10, -3)

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 7

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(2, 7)

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 9

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 9)

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -9

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

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

ORIGINAL positive integer absolute magnitude=10>1

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

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

upf extract divisor2 occurrence0 = 5

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

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 10

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

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

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

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

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

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 1

Mobius inversion ORIGINAL inverse F(n/10):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/10) f(1) ORIGINAL a times divisor := mul(5, 1)

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 5

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(5, -3)

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

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

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 2

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 2)

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 2

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(78, -24, -9, 2)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 47

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(5, 10)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 50

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(50, -3)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 47

Mobius inversion ORIGINAL inverse:actual n=10;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]]

complete reconstructed=47;direct f(n)=47;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

逐步反向核對

• S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S060 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct plus ORIGINAL b，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 direct ORIGINAL a times n，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 complete inverse-term sum，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10)，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/10) full forward divisor sum，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/10) f(1) 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/10) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S053 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S052 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10，不以小數近似替代。
• S051 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S050 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S049 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S046 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，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 Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5)，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5，不以小數近似替代。
• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · 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。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · 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。
• S028 · 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。
• S027 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S025 · 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。
• S024 · 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。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · 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。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · 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。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · 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。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, 是)

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。