slug: oeis-a350900-schulte-gcd-quotient-row-sum-convolution bibkey: schulte2022a350900 doi: null url: https://oeis.org/A350900 triage: theorem motivation_gids:

• D5/S3/ArithSums/SchulteGcdQuotientRowSumConvolution

Schulte’s gcd-quotient row-sum convolution

Problem

OEIS A350900, NAME (%N, verbatim):

Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

The settled FORMULA (%F, verbatim) is:

Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.

The full-quantifier reading is: for every positive natural n, define rowSum(n) = Sum_{i=1..n} Sum_{k=1..n} gcd(i,n)/gcd(gcd(i,k),n); then

rowSum(n) = Sum_{d|n} d*phi(d) * (Sum_{e|(n/d)} e*phi(e)).

Here phi is Euler’s totient, A002618(n) is n*phi(n), and A057660(n) is Sum_{d|n} d*phi(d). The identity gcd(gcd(i,k),n) = gcd(gcd(i,n),k) shows the denominator of each row summand divides its numerator; these are exact natural quotients. For each d|n the complementary quotient n/d is also exact. Only the quoted row-sum formula is settled. The %C conjecture for an arbitrary arithmetic function f is outside this claim.

Motivation

The two-level gcd quotient reduces, for fixed i, to a periodic sum in k whose divisor fibres have totient cardinalities. Applying the same totient count to the outer gcd fibres exposes the two convolution factors.

Gap

OEIS A350900 continues to mark the row-sum formula as a conjecture and gives no proof. 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 (https://oeis.org/history?seq=A373059):

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

The equivalence follows from the A057660 formula sigma_2(m^2)/sigma(m^2) = Sum_{e|m} e*phi(e). The A373059 formula is prior documentation of the equivalent identity. This dossier makes no first-discovery, first-resolution, or historical-priority claim.

Tóth’s 2010 gcd-sum survey records Cesàro’s general fibre formula Sum_{k=1..n} f(gcd(k,n)) = Sum_{d|n} f(d)*phi(n/d) and the A057660 divisor sum. These general results supply the established background for the proof, but the survey does not state the nested A350900 identity.

As of September 16, 2026, the complete inspected A350900 history (13 revisions), the complete inspected A373059 history (19 revisions), Tóth’s 2010 survey (https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html), arXiv:2110.07271, arXiv:1306.1020, and Bordellès’s 2015 Journal of Integer Sequences paper (https://cs.uwaterloo.ca/journals/JIS/VOL18/Bordelles/bord21.pdf) yielded no complete prior proof of the target identity. The literature result is not-found-in-searched-scope, never a claim that the proof is globally novel.

No equal or stronger theorem was found in the current repository or pinned Mathlib. The repository’s other gcd-sum results concern different summands, and pinned Mathlib supplies the totient, divisor-reindexing, periodic-gcd, and finite-fibre lemmas used by the proof rather than the target identity.

Route

1. Rewrite gcd(gcd(i,k),n) as gcd(gcd(i,n),k) and rotate the 1..n sums to complete residue ranges; endpoint equality follows from gcd periodicity.
2. With g=gcd(i,n), a local calculation inside result establishes Sum_{k in range(n)} g/gcd(g,k) = (n/g)*Sum_{f|g} f*phi(f). Periodicity splits n into n/g blocks of length g. Grouping a block’s k by gcd(g,k) gives totient cardinalities for the complementary divisors; reindexing produces the divisor sum.
3. Stratify the outer i by gcd(n,i)=g, giving phi(n/g) indices for each g|n. Reindex the complementary divisors by d=n/g to obtain the inlined A057660 convolution factor.

The public result has proof_shape: bind-only, no escape witness, and admission_basis: open-problem-resolution. It is the sole public theorem in this module and settles exactly the quoted named conjecture; the admission classification is not a claim to the first proof or first resolution. Registration issue #8240 was created at 2026-09-16T01:17:03Z, before the first proof probe at 2026-09-16T01:18:31.892Z. The proof directly reuses pinned Mathlib statements through instantiation, finite-fibre regrouping, divisor reindexing, and normalization. There is no direct frozen D5 dependency; all supporting facts remain local have terms inside result.

utility: none applies because rowSum is a symbolic definition and result is an unbounded symbolic identity, not a bounded enumeration, checker, numeric reduction, or certified finite instance.

Falsifier

Any positive natural n with unequal sides of the displayed convolution would refute the result. An i,k in 1..n for which gcd(gcd(i,k),n) fails to divide gcd(i,n) would refute the claimed exactness of a summand.

Evidence

The Lean definition rowSum transcribes the A350900 double sum with natural division. The theorem result retains the hypothesis 0 < n and proves only the displayed divisor convolution. Its proof establishes the fixed-gcd fibre identity locally for every outer index and then groups the outer indices by their gcd with n.

The proof uses the standard three permitted axioms through Mathlib. Its direct imports supply the totient/divisor identities and interval-sum conversion. No statement about an arbitrary arithmetic function occurs in the public surface.

Triage

theorem; the formal target is exactly the quoted %F row-sum formula over all positive natural row indices.

ASSUMED-UNVERIFIED

The literature search is not a proof of historical novelty. 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. A complete prior proof outside the searched scope remains ASSUMED-UNVERIFIED. No priority claim is made. The arbitrary-function conjecture in A350900 is outside the formal statement and remains unresolved by this result.