Metadata
bibkey: hanna2013a226838
authors: Paul D. Hanna
year: 2013
title: "OEIS A226838 — E.g.f.: exp( Sum_{n>=1} x^(n^2) / n^n )."
doi: null
url: https://oeis.org/A226838
claim: "a(n) == 1 (mod 6) (conjecture - valid up to n=1024; if true for n>=0, why?)."
strata_touched:
- D5/S1/Recurrence/Residue/HannaA226838Congruence
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A226838
Revision #14 in the public history: https://oeis.org/history?seq=A226838 The entry is by Paul D. Hanna, June 19, 2013; revision #14 is stamped June 20, 2013 21:05:34. This OEIS entry has no DOI, source arXiv identifier, or page number.
Source statement
The revision #14 NAME field reads verbatim:
E.g.f.: exp( Sum_{n>=1} x^(n^2) / n^n ).
The FORMULA conjecture reads verbatim:
a(n) == 1 (mod 6) (conjecture - valid up to n=1024; if true for n>=0, why?).
Over rational formal power series, let
H(X)=sum_{m>=1} X^(m^2)/m^m and A(X)=exp(H(X)).
The sequence term is a(n)=n!*[X^n]A(X). The square exponent and power
denominator are literal source data. The all-index conclusion includes zero
and is expressed by an integer witness k with a(n)=1+6*k in the rationals;
integrality is part of the conclusion rather than a premise.
The source’s reported check through 1024 is finite evidence, not an all-index proof.
Mathematical scope
The repository-derived proof identifies the factorial-normalized inner
coefficients with natural factorial quotients. For every m>=2,
6*m^m divides (m^2)!, so this integral inner sequence reduces modulo six
to the identity sequence. The existing integral EGF composition law transfers
that reduction to the outer exponential. The resulting integer divisibility
is cast back into the exact rational equality. This proof is not attributed
to the source entry. The argument is unbounded and symbolic.
The source and bounded prior-work qualification are attached to issue #23866, including its immutable preregistration 6507420858. No full answer was located in the completed bounded search scopes; that result does not establish literature-wide absence or priority. The weighted-permutation, Hurwitz-series and EGF-composition papers considered there supply related machinery or conditional conclusions, not an identified literal answer to this event. An independent focused reading of Yuta Nishibuchi, A Unified Framework for p-adic Congruences and p-adic Interpolations of Sequences, arXiv:2610.11648v2, established no exact implication for this series in that source. The checked records are the history, v2 abstract, HTML, PDF, and source archive. The paper supplies conditional p-adic criteria, but applying them here would require target-specific normalized coefficient valuations or a valid decomposition, the correct period, prime strength, and base-value links for both p=2 and p=3, all-index coverage including zero, and a global integrality bridge to an integer witness. These conditions remain unverified for this series. This reading does not claim that the conditions are false or rule out the method, and it does not establish a global novelty or priority claim. The paper is prior-work context, not the identifier of the OEIS source and not an asserted proof of this conjecture.