bibkey: hanna2009logarithmicclosure
authors: Paul D. Hanna
year: 2009
title: "Hanna's logarithmic integer-series power closure conjecture"
doi: null
url: https://oeis.org/A167138
claim: "Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * 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/HannaLogarithmicPowerClosure
license: citation-only
triage: anchor

Hanna’s logarithmic closure question

The quoted general conjecture occurs identically in OEIS A167138 and A167139. A167141 repeats the same assertion with capitalization and grammar differences. These are locations of one question. The partition example A167137 and the square examples do not restrict its arbitrary sequence L. The A167138/A167139 AUTHOR attribution is Paul D. Hanna, Nov 03 2009. A167141 attributes it to Hanna, Nov 04 2009, and prints:

CONJECTURE: For all integer m>0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

Its immutable source SHA-256 is 05a1d290827a2932b439b91520c4ecd907f1b0275f50331af0a0ebbb0396a475. The retained revision-one histories already contain the assertion.

A167138 and A167139 revision 13 use ordinary coefficients: their examples are ordinary integer series, and the signed first-kind Stirling transform is divided by the ordinary degree factorial. Every natural sequence index and degree, including zero, is retained. The field interpretation uses any characteristic-zero field and its canonical integer image. No convergence, positivity, normalization or support restriction is needed.

The immutable OEIS mirror at bd331f23038f1570c996386c6c221342aeb36fc5 has source hashes 45abbe815cbec381fd5b71c3a2679a2981442ce6091ae8594ea5a0cb4463c255 for A167138 and bcb45b2a17e9ecb8502ec5f54614d17092d97397935b30607171e8f9aeef4b82 for A167139. Preregistration is issue https://github.com/the-omega-institute/trureturing/issues/16500.

Classical suppliers

Boyadzhiev and Frontczak, The Hadamard product of series with Stirling numbers of the second kind and other special numbers, Electronic Journal of Mathematics 4 (2022), 32–38, DOI 10.47443/ejm.2022.024, Corollary 3.1, supply the fixed-column Hadamard EGF identity. Their analytic theorem is used only through its finite entire-column identity. Arbitrary formal input series need no analytic hypothesis. The finite formal extension uses factorial-weighted Stirling inclusion-exclusion and degreewise finite sums. Classical supplier identities are not claimed as new.

The exp/log inverse leaves are adapted from Ralf Stephan’s Mathlib Mathlib/RingTheory/PowerSeries/Log.lean at immutable revision ff5815e8fe1ecf6880c38ea997fb56395951822b, source SHA-256 ced40d5066d36c16787715c93fbadb9c3b63b773813ebe7c4f5bfa9ee21bfdad. Copyright (c) 2026 Ralf Stephan; Apache-2.0. The full license is docs/reports/log-inverse-suppliers/mathlib-LICENSE.txt, SHA-256 b40930bbcf80744c86c46a12bc9da056641d722716c378f5659b9e555ef833e1. The complete immutable upstream tree has no NOTICE path. Upstream Lean 4.35 differs from the adopted Lean 4.33 pin; dependency form is unavailable. Only consumed inverse leaves are adapted. Retire them when this repository’s adopted Mathlib Log.lean contains PowerSeries.subst_exp_log and PowerSeries.subst_log_exp_sub_one with equivalent compiled types; direct upstream calls then replace the adaptation.

Bounded literature status

The original source qualification inspected the full primary 2022 paper, current and immutable OEIS records and retained histories, exact-ID Crossref/arXiv/MSE/MO queries, related Stirling/Hadamard queries and bounded all-state repository ownership bodies. No exact earlier settlement was identified in that scope. SeqFan archive access failed, and several general search services rejected or rate-limited requests. This is neither worldwide open-status certification nor a proof-novelty claim. A later exact settlement would require reuse and would exclude a new-question count.

Verified locator

https://oeis.org/A167138