第 21749 題：Möbius函數

原題與已知條件

positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={"n":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={"n":24}。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-function

逐步推導

source operation=mobius-divisor-sum;original snapshot and mathematics retained

explicit ordered source output fields=[n, divisor_mu_rows, sum, target, equal]

ORIGINAL n=24;complete positive candidate range1..n

ALL actual positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24]

mobius divisor1 of original n=24:actual positive input for mu=1

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

mobius divisor1 of original n=24:input=1;COMPLETE prime multiset=[];distinct primes=[]

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

mobius divisor2 of original n=24: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 divisor2 of original n=24:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

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

mobius divisor3 of original n=24: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 divisor3 of original n=24:input=3;COMPLETE prime multiset=[3];distinct primes=[3]

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

mobius divisor4 of original n=24:actual positive input for mu=4

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

ORIGINAL positive integer absolute magnitude=4>1

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

upf extract divisor2 occurrence0 := div(4, 2)

upf extract divisor2 occurrence0 = 2

upf extract divisor2 occurrence1 := div(2, 2)

upf extract divisor2 occurrence1 = 1

next divisor=2;previous remaining=4;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

mobius divisor4 of original n=24:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

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

mobius divisor6 of original n=24: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 divisor6 of original n=24:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]

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

mobius divisor8 of original n=24:actual positive input for mu=8

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

ORIGINAL positive integer absolute magnitude=8>1

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

upf extract divisor2 occurrence0 := div(8, 2)

upf extract divisor2 occurrence0 = 4

upf extract divisor2 occurrence1 := div(4, 2)

upf extract divisor2 occurrence1 = 2

upf extract divisor2 occurrence2 := div(2, 2)

upf extract divisor2 occurrence2 = 1

next divisor=2;previous remaining=8;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=8

complete increasing prime multiset=[2, 2, 2];full product=8;a composite terminal remainder would have an already tested smaller divisor

mobius divisor8 of original n=24:input=8;COMPLETE prime multiset=[2, 2, 2];distinct primes=[2]

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

mobius divisor12 of original n=24:actual positive input for mu=12

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

ORIGINAL positive integer absolute magnitude=12>1

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

upf extract divisor2 occurrence0 := div(12, 2)

upf extract divisor2 occurrence0 = 6

upf extract divisor2 occurrence1 := div(6, 2)

upf extract divisor2 occurrence1 = 3

next divisor=2;previous remaining=12;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 3)

upf canonical complete product factor2 = 12

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=12

complete increasing prime multiset=[2, 2, 3];full product=12;a composite terminal remainder would have an already tested smaller divisor

mobius divisor12 of original n=24:input=12;COMPLETE prime multiset=[2, 2, 3];distinct primes=[2, 3]

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

mobius divisor24 of original n=24:actual positive input for mu=24

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

ORIGINAL positive integer absolute magnitude=24>1

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

upf extract divisor2 occurrence0 := div(24, 2)

upf extract divisor2 occurrence0 = 12

upf extract divisor2 occurrence1 := div(12, 2)

upf extract divisor2 occurrence1 = 6

upf extract divisor2 occurrence2 := div(6, 2)

upf extract divisor2 occurrence2 = 3

next divisor=2;previous remaining=24;all smaller divisors exhausted

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

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

upf canonical complete product factor0 = 2

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

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 3)

upf canonical complete product factor3 = 24

exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=24

complete increasing prime multiset=[2, 2, 2, 3];full product=24;a composite terminal remainder would have an already tested smaller divisor

mobius divisor24 of original n=24:input=24;COMPLETE prime multiset=[2, 2, 2, 3];distinct primes=[2, 3]

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

Mobius complete divisor mu sum := sum(1, -1, -1, 0, 1, 0, 0, 0)

Mobius complete divisor mu sum = 0

ORIGINAL n=24;ALL increasing [divisor,mu] rows=[[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]]

exact sum=0;target=0;squarefree divisors correspond to subsets of DISTINCT primes,so sum=(-1)^subset_size summed over all subsets=(1-1)^r;for n1,r0 gives1

逐步反向核對

• 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 complete divisor mu sum，不以小數近似替代。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S057 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• 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 occurrence1，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 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 upf canonical complete product factor2，不以小數近似替代。
• S039 · 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。
• S038 · 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。
• 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 upf extract divisor2 occurrence2，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 upf canonical complete product factor1，不以小數近似替代。
• S028 · 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。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · 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。
• S020 · 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。
• 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 — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，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 upf canonical complete product factor0，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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

(24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, 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。