Question 22151: Keep mu(d) Paired with F(n Divided by d)

Original question and givens

Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

• G001 — Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-mobius-inversion-finite-divisors

Step-by-step derivation

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=1;check complete integer range1..1

ALL positive divisors INCREASING=[1];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=Yes;repeated prime=No

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(1, 1)

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 1

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

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

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(1, -12)

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = -11

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(-11)

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = -11

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, -11)

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = -11

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(-11)

Mobius inversion ORIGINAL inverse complete inverse-term sum = -11

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(1, 1)

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 1

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(1, -12)

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = -11

Mobius inversion ORIGINAL inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, -11, -11]]

complete reconstructed=-11;direct f(n)=-11;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1, [[1, 1, 1, -11, -11]], -11, -11, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.