第 21798 題：Möbius函數

原題與已知條件

positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={"n":166,"factors":[2,83]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={"n":166,"factors":[2,83]}。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-factorization-audit;original snapshot and mathematics retained

explicit ordered source output fields=[n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate]

mobius ORIGINAL factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[]

ALL [divisor,remainder] rows=[];prime=是

Mobius candidate exact full product at factor0 := mul(1, 2)

Mobius candidate exact full product at factor0 = 2

mobius ORIGINAL factor index1:ORIGINAL candidate factor=83;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4, 5, 6, 7, 8, 9]

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 3], [6, 5], [7, 6], [8, 3], [9, 2]];prime=是

Mobius candidate exact full product at factor1 := mul(2, 83)

Mobius candidate exact full product at factor1 = 166

mobius candidate canonical comparison:actual positive input for mu=166

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

ORIGINAL positive integer absolute magnitude=166>1

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

upf extract divisor2 occurrence0 := div(166, 2)

upf extract divisor2 occurrence0 = 83

next divisor=2;previous remaining=166;all smaller divisors exhausted

extracted multiplicity=1;next remaining=83;remaining modulo2=1;extracted multiset=[2]

next divisor=3;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo3=2;extracted multiset=[2]

next divisor=4;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo4=3;extracted multiset=[2]

next divisor=5;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo5=3;extracted multiset=[2]

next divisor=6;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo6=5;extracted multiset=[2]

next divisor=7;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo7=6;extracted multiset=[2]

next divisor=8;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo8=3;extracted multiset=[2]

next divisor=9;previous remaining=83;all smaller divisors exhausted

extracted multiplicity=0;next remaining=83;remaining modulo9=2;extracted multiset=[2]

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

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 83)

upf canonical complete product factor1 = 166

exhausted trial divisors below10;terminal remaining=83;d*d>remaining;ORIGINAL n=166

complete increasing prime multiset=[2, 83];full product=166;a composite terminal remainder would have an already tested smaller divisor

mobius candidate canonical comparison:input=166;COMPLETE prime multiset=[2, 83];distinct primes=[2, 83]

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

ORIGINAL n=166;ORIGINAL factor order=[2, 83];ALL primality flags=[是, 是];EXACT full product=166

valid prime product=是;candidate mu_or_null=1;canonical mu=1;agreement=是;large product is NOT a rounded Number output field

逐步反向核對

• 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 — 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 canonical complete product factor1，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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 — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S007 · 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。
• 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 candidate exact full product at factor1，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 — 以精確有理數重算 Mobius candidate exact full product at factor0，不以小數近似替代。
• 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

(166, [2, 83], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)

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。