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bibkey: schulte2022a350900 authors: Werner Schulte year: 2022 title: “OEIS A350900, T(n,k) = Sum_{i=1..n} gcd(i,n)/gcd(gcd(i,k),n), with the conjectured row sums (n phi(n)) * (Sum_{d|n} d phi(d))” doi: null url: https://oeis.org/A350900 claim: “%N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n. %F Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.” strata_touched:

  • D5/S3/ArithSums/SchulteGcdQuotientRowSumConvolution license: citation-only triage: anchor

OEIS A350900

Werner Schulte’s row formula defines exact natural-number gcd quotients. The row-sum statement identifies their sum as a Dirichlet convolution: A002618(n) is n * phi(n), and A057660(n) is Sum_{d|n} d * phi(d). The separate conjecture for an arbitrary arithmetic function is outside the result’s scope.

OEIS A373059 identifies the same double sum as the row sums of A350900. Seiichi Manyama recorded the equivalent formula in revision 11 on May 24, 2024, at 11:06:04 EDT:

Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).

OEIS A057660 gives sigma_2(d^2)/sigma(d^2) = Sum_{e|d} e * phi(e), so this is the same convolution formula, although A350900 continues to label its row-sum formula as a conjecture. The A373059 formula is prior documentation of the equivalent identity; no first-discovery or first-resolution claim is made here.

László Tóth’s survey A Survey of Gcd-Sum Functions, Journal of Integer Sequences 13 (2010), Article 10.8.1, records Cesàro’s general formula

Sum_{k=1..n} f(gcd(k,n)) = Sum_{d|n} f(d) * phi(n/d)

for an arbitrary arithmetic function f, and records A057660 as the sum of the orders of elements of a cyclic group. This general fibre-counting theorem explains the proof mechanism, but the survey does not state the nested A350900 row-sum identity.

Literature status as of September 16, 2026

The complete inspected histories of A350900 (13 revisions) and A373059 (19 revisions), together with Tóth’s 2010 survey, arXiv:2110.07271, arXiv:1306.1020, and Bordellès’s 2015 Journal of Integer Sequences paper, yielded no complete prior proof of the A350900 row-sum identity. The result is not-found-in-searched-scope, not a claim that the proof is globally novel.

Material limits are that A057660 revisions 63 through 1 were not inspected; the Gould-Shonhiwa 1997 full texts, Tóth’s 1998 regular-convolution paper, some cyclic-group sources, and the U338 solution were not obtained.

Verified locator

  • URL: https://oeis.org/A350900
%N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.
%F Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.
%A _Werner Schulte_

Related verified locators:

  • OEIS A373059: https://oeis.org/A373059
  • A373059 history: https://oeis.org/history?seq=A373059
  • OEIS A057660: https://oeis.org/A057660
  • Tóth’s survey: https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html
  • arXiv:2110.07271: https://arxiv.org/abs/2110.07271
  • arXiv:1306.1020: https://arxiv.org/abs/1306.1020
  • Bordellès, JIS 2015: https://cs.uwaterloo.ca/journals/JIS/VOL18/Bordelles/bord21.pdf

The checked sources support a formal proof of the A350900 formula, not a claim of historical priority. A complete prior proof outside the searched scope remains ASSUMED-UNVERIFIED.