Metadata
bibkey: hanna2009a156216
authors: Paul D. Hanna
year: 2009
title: "OEIS A156216: diagonal ghost-power integrality conjecture"
doi: null
url: https://oeis.org/A156216/internal
claim: "More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture)."
strata_touched:
- D5/S1/Recurrence/Witt/HannaDiagonalGhostPowerIntegrality
license: citation-only
triage: anchor
Verified locator
DOI: null
Source: https://oeis.org/A156216/internal
The source is the second %C comment of OEIS A156216, internal revision
#3 Mar 30 2012 18:37:16, attributed to Paul D. Hanna, February 6, 2009.
It is unpaginated. No arXiv source of the conjecture is identified.
More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture).
The expression is an ordinary formal power series. Its constant coefficient is one. The sequence takes values in an arbitrary characteristic-zero field; “integer coefficients” means the canonical image of the integers. Index zero is unused. There is no positivity or analytic convergence condition. The Lucas-number example in the entry does not restrict the universal comment.
The proof extracts the integer Euler ledger of the input, obtains its divisor ghosts, preserves their local Gauss congruences under diagonal self-powers, and reconstructs the output by integer Moebius inversion and formal differential uniqueness. The classical Dold/Euler bridge is discussed in Byszewski, Graff and Ward, Dold sequences, periodic points, and dynamics, arXiv:2007.04031v4, DOI 10.1112/blms.12531. This supporting paper is not the source of Hanna’s conjecture.
The experiment is in the-omega-institute/trureturing-experiments, commit
8e7f1dcf75f89aa584f0667e326ea30a33ea09a0, at
docs/reports/oeis-a156216-hanna-diagonal-power-integrality/check.py, SHA-256
4a8078da8433455f65e9d9d52a52e0b1d35ee98ae09611d5d098317826a7f1f9.
The recorded command is python3 docs/reports/oeis-a156216-hanna-diagonal-power-integrality/check.py;
its exit code is 0 and final line is ALL_CHECKS_OK. These bounded readings
are not the proof of the universal assertion.
The bounded literature screen in preregistration issue
https://github.com/the-omega-institute/trureturing/issues/21407
found no settlement in the inspected sources. Worldwide priority and
literature completeness remain ASSUMED-UNVERIFIED; the inaccessible
Du-Huang-Li 2003 paper and Newton-Euler paper remain reading boundaries.