第 22163 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":13,"a":3,"b":0}。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":13,"a":3,"b":0}。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=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 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(13, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 13

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(3, 0)

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

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor := mul(3, 13)

Mobius inversion ORIGINAL inverse F(n/1) f(13) ORIGINAL a times divisor = 39

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b := sum(39, 0)

Mobius inversion ORIGINAL inverse F(n/1) f(13) plus ORIGINAL b = 39

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(3, 39)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 42

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 42)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 42

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(13, 13)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor13 = 1

Mobius inversion ORIGINAL inverse F(n/13):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/13) f(1) ORIGINAL a times divisor := mul(3, 1)

Mobius inversion ORIGINAL inverse F(n/13) f(1) ORIGINAL a times divisor = 3

Mobius inversion ORIGINAL inverse F(n/13) f(1) plus ORIGINAL b := sum(3, 0)

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

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum := sum(3)

Mobius inversion ORIGINAL inverse F(n/13) full forward divisor sum = 3

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) := mul(-1, 3)

Mobius inversion ORIGINAL inverse mu(13) times SAME F(n/13) = -3

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(42, -3)

Mobius inversion ORIGINAL inverse complete inverse-term sum = 39

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(3, 13)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 39

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(39, 0)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 39

Mobius inversion ORIGINAL inverse:actual n=13;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]]

complete reconstructed=39;direct f(n)=39;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

• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 Mobius inversion ORIGINAL inverse direct 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 direct ORIGINAL a times n，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 complete inverse-term sum，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 mu(13) times SAME F(n/13)，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/13) full forward divisor sum，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/13) f(1) plus ORIGINAL b，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/13) f(1) ORIGINAL a times divisor，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 — 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。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• 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

(13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, 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。