bibkey: oeis2026a400585
authors: Jishnu Babu Ranitha; Vaclav Kotesovec
year: 2026
title: A400585 — odd-cofactor binomial divisor sums and a cubic summatory conjecture
doi: null
url: https://oeis.org/A400585
claim: Revision 22 defines the odd-cofactor divisor sum and conjectures its summatory asymptotic 7*zeta(3)*n^3/48.
strata_touched:
  - D5/S3/ArithSums/A400585OddCofactorAsymptotic
license: citation-only
triage: anchor

The A400585 summatory conjecture

Verified locator

https://oeis.org/A400585

The internal entry is https://oeis.org/A400585/internal. The machine-readable entry is https://oeis.org/search?q=id:A400585&fmt=json, revision 22.

Statement

The NAME reads:

a(n) = Sum_{d|n, n/d odd} binomial(d, 2).

The FORMULA conjecture reads:

Conjecture: Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/48. - Vaclav Kotesovec, Oct 01 2026

The offset is one. Divisors are positive and the cofactor quotient is exact. Here zeta(3) means the convergent real series over positive integers of their reciprocal cubes; the defining zeta series is also given by https://dlmf.nist.gov/25.2.E1. The source provides the conjecture, not a proof.

Scope

The Lean source defines the natural divisor sum a, the real positive-prefix sum S, and the real cubic reciprocal series Z3. Its sole theorem gives the ratio limit along all natural indices, with coefficient exactly 7*Z3/48 and limit one. It introduces no parity, primality or finite-range restriction. The proof uses divisor/cofactor reindexing, the finite binomial sum, the floor-ratio limit and dominated-series convergence. The derived result does not assert a literature priority claim. The generating-function formula and the separate sigma-function formula in the entry are outside this theorem.

Attribution and license

The entry credits Jishnu Babu Ranitha for the sequence and Vaclav Kotesovec for the conjecture. OEIS content is licensed under CC BY-SA 4.0; see https://oeis.org/wiki/The_OEIS_End-User_License_Agreement. The quotations above retain their source attribution. No upstream proof code is reproduced.