Question 22121: The Forward Transform Sums over All Positive Divisors

Original question and givens

Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. 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, divisor_value_rows, F_n, certificate).

• G001 — Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. 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, divisor_value_rows, F_n, certificate).

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-forward-divisor-transform;original snapshot and mathematics retained

explicit ordered source output fields=[n, divisor_value_rows, F_n, certificate]

Mobius inversion ORIGINAL forward:actual positive input=21;check complete integer range1..21

ALL positive divisors INCREASING=[1, 3, 7, 21];not only primes/proper divisors or all integers<=n

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor := mul(1, 1)

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor = 1

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b := sum(1, 8)

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b = 9

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor := mul(1, 3)

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor = 3

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b := sum(3, 8)

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b = 11

Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor := mul(1, 7)

Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor = 7

Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b := sum(7, 8)

Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b = 15

Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor := mul(1, 21)

Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor = 21

Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b := sum(21, 8)

Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b = 29

Mobius inversion ORIGINAL forward full forward divisor sum := sum(9, 11, 15, 29)

Mobius inversion ORIGINAL forward full forward divisor sum = 64

Reverse checks

• S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• 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

(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)

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.