bibkey: spezia2018a320043
authors: Stefano Spezia; OEIS Foundation Inc.
year: 2018
title: "OEIS A320043: Row sums of the triangle A322550"
doi: null
url: https://oeis.org/A320043/internal
claim: "Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96."
strata_touched:
  - D5/S3/ArithSums/SpeziaA320043SquareValuesRefutation
license: citation-only
triage: anchor

Spezia’s A320043 square-value conjecture

Verified locator

Source: https://oeis.org/A320043/internal

OEIS A320043, revision 43, NAME, COMMENTS, FORMULA and OFFSET fields. This source is unpaginated and has no associated arXiv identifier or DOI located in the literature screen recorded in issue #24212.

NAME:

Row sums of the triangle A322550.

The sole COMMENTS conjecture:

Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.

The first FORMULA:

a(n) = Sum_{k=1..n} (n + 1 - k)^2*k/gcd(n + 1 - k, k)^3.

OFFSET is “1,2”; the sequence domain begins at n = 1. In the formal expression, n and k are natural numbers and the sum is over Finset.Icc 1 n. The selected summands are exact natural quotients: if d divides u and v, then d³ divides u²v. Natural subtraction agrees with ordinary subtraction on this interval. The extension at n = 0 is an empty sum and is excluded from the conjecture. Perfect squares are represented by m² with m natural.

The refutation gives the additional index 708158976, with square root 144767444036350576. A prime-row identity and the integer Pell equation p² − 12y² = 1 yield this value without enumerating the large row. The smallest additional square index, infinitude, and a complete classification remain open. The source’s other displayed formulas do not depend on the conjecture.

The literature screen in #24212 found no settlement in its inspected sources. Completeness and novelty remain ASSUMED-UNVERIFIED.