Question 22116: The Forward Transform Sums over All Positive Divisors

Original question and givens

Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. 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=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. 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=16;check complete integer range1..16

ALL positive divisors INCREASING=[1, 2, 4, 8, 16];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, 3)

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b = 4

Mobius inversion ORIGINAL forward f(2) ORIGINAL a times divisor := mul(1, 2)

Mobius inversion ORIGINAL forward f(2) ORIGINAL a times divisor = 2

Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b := sum(2, 3)

Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b = 5

Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor := mul(1, 4)

Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor = 4

Mobius inversion ORIGINAL forward f(4) plus ORIGINAL b := sum(4, 3)

Mobius inversion ORIGINAL forward f(4) plus ORIGINAL b = 7

Mobius inversion ORIGINAL forward f(8) ORIGINAL a times divisor := mul(1, 8)

Mobius inversion ORIGINAL forward f(8) ORIGINAL a times divisor = 8

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

Mobius inversion ORIGINAL forward f(8) plus ORIGINAL b = 11

Mobius inversion ORIGINAL forward f(16) ORIGINAL a times divisor := mul(1, 16)

Mobius inversion ORIGINAL forward f(16) ORIGINAL a times divisor = 16

Mobius inversion ORIGINAL forward f(16) plus ORIGINAL b := sum(16, 3)

Mobius inversion ORIGINAL forward f(16) plus ORIGINAL b = 19

Mobius inversion ORIGINAL forward full forward divisor sum := sum(4, 5, 7, 11, 19)

Mobius inversion ORIGINAL forward full forward divisor sum = 46

Reverse checks

• S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(16) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(16) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(2) 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

(16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, 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.