Metadata
bibkey: hanna2025a382124a382125
authors: Paul D. Hanna
year: 2025
title: OEIS A382124 and A382125, sigma-product exponential series
doi: null
url: https://oeis.org/A382124
claim: "Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0. Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0."
strata_touched:
- D5/S3/Arith/Robin/HannaSigmaProductExpModThree
- D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A382124
Companion source: https://oeis.org/A382125
Statements
A382124, NAME (unpaginated):
G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
A382124, COMMENT (unpaginated):
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.
A382125, NAME (unpaginated):
G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
A382125, COMMENT (unpaginated):
Conjectures: a(3n) == A382124(n) (mod 3) for n >= 0; a(3n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.
The coefficient index begins at zero. Formal substitution into the rational exponential series expresses each generating function; the logarithmic input has constant coefficient zero. Integer witnesses express the integer congruences. The proof obtains integrality from the Dold congruences and then uses cubing in characteristic three. No analytic convergence is required.