bibkey: hanna2009affineclosure
authors: Paul D. Hanna
year: 2009
title: "Hanna's logarithmic integer-series arithmetic-subsequence conjecture"
doi: null
url: https://oeis.org/A167540
claim: "For integers m>0, b>=0, Sum_{n>=0} L(m*n+b) * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series."
strata_touched:
  - D5/S1/Recurrence/Algebraic/HannaLogarithmicAffineClosure
license: citation-only
triage: anchor

Hanna’s arithmetic-subsequence question

The OEIS CONJECTURE comment attributes the exact quoted assertion to Paul D. Hanna, Nov 06 2009. The initial retained official history already contains it; the subsequent retained revision changes attribution formatting. The immutable mirror is seq/A167/A167540.seq at OEIS data commit 72c53dd50967ac7166f7d31bf79be95174ec5bc6, SHA-256 97049e84b0bc33cbccb54c92588ee3472280c6236d0d6e8517ef14c8e716e9a6. The entry’s displayed ordinary generating function and coefficient extraction code fix “integer series” to mean integer ordinary coefficients. Its named example uses log(1-2x), whereas the universal conjecture explicitly uses log(1+x); the example does not alter that logarithm or restrict arbitrary L. A167532 is a pointwise-power question, distinct from arithmetic extraction.

Classical suppliers

DLMF 26.8, equations E6, E8, E10 and E12, records the classical Stirling inclusion-exclusion, logarithmic, falling-factorial and exponential column identities. Rahmani, Generalized Stirling transform, arXiv 1212.0957v1, DOI 10.18514/mmn.2014.1084, and Bounebirat–Laissaoui–Rahmani, Some combinatorial identities via Stirling transform, DOI 10.7546/nntdm.2018.24.4.92-98, supply related inversion and shifted-index identities. Their inspected complete statements do not assert the arbitrary dilation question quoted here. These are citation sources, not copied code.

The formal proof directly uses the repository’s factorial-weighted stirling2_inclusion_exclusion and original HannaLogarithmicPowerClosure.reconstruction, subst_exp_log, map_transform, coeff_transform and T_window. The original inverse leaves retain their Apache-2.0 source and full-license provenance in hanna2009logarithmicclosure.md; no inverse proof is copied into the affine module. The pinned Mathlib exponential-rescaling and coefficient identities are applied directly. No classical supplier or proof method is claimed novel.

Bounded literature status

The source qualification inspected current and immutable OEIS records and retained history, exact-ID arXiv/MSE/MO/GitHub queries, related Stirling and integer-valued-polynomial searches, the complete Rahmani source and the complete seven-page NNTDM paper, and bounded repository issue/PR ownership indices. No exact whole settlement or conflicting owner was identified within those scopes. Blocked history views, rate-limited queries, unavailable SeqFan and challenged general search services provide no absence evidence. There is no global absence, priority or method-novelty claim. Preregistration and the full-field interpretation are at https://github.com/the-omega-institute/trureturing/issues/16545.

Verified locator

https://oeis.org/A167540

https://raw.githubusercontent.com/oeis/oeisdata/72c53dd50967ac7166f7d31bf79be95174ec5bc6/seq/A167/A167540.seq

https://dlmf.nist.gov/26.8

https://arxiv.org/abs/1212.0957

https://doi.org/10.18514/mmn.2014.1084

https://doi.org/10.7546/nntdm.2018.24.4.92-98