trureturing — Discovering truth

bibkey: bala2025a258899a258903
authors: Paul D. Hanna; Peter Bala
year: 2025
title: "OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures"
doi: null
url: https://oeis.org/A258899
claim: "For each of A258899 and A258903: Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k)."
strata_touched:
  - D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A258899

Source: https://oeis.org/A258903

The entries are not paginated. The locators are the NAME, COMMENTS, FORMULA, OFFSET and initial-term fields of the internal entries: https://oeis.org/A258899/internal and https://oeis.org/A258903/internal. A258899 revision #17 and A258903 revision #12 are both dated 2025-08-09. Hanna introduced the sequences in 2015; Bala’s conjecture comments are dated 2025-08-03 and 2025-08-08, respectively. No source DOI or arXiv identifier applies.

Source statements

Both COMMENTS fields contain the same conjecture sentence, quoted verbatim:

Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k).

A258899 NAME:

E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

A258899 FORMULA:

For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!).

A258903 NAME:

E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(3*x^n) / n!.

A258903 FORMULA:

For n >= 1, a(n) = Sum_{d divides n} 2^d * 3^(n/d) * n!/(d!*(n/d)!).

Both entries have offset zero and a(0)=1. The A258903 COMMENTS example modulo 7 begins at n=1; it does not change the value at index zero.

Mathematical interpretation

The Lean definitions use the two finite FORMULA expressions with the stated initial value. The natural divisor quotient is exact on positive divisors. The factorial denominator is positive. For a factorization n=de with d,e>=2, d+e<=n and factorial divisibility gives d!e! dividing n!; at each endpoint the quotient is 1. Natural-number division therefore agrees with the integer quotient appearing in the source formulas.

The two assertions independently quantify a positive natural modulus m, an onset N, a positive period T dividing Nat.totient m, and all n>=N. The module proves both with N=2m+4 and T=Nat.totient m, including m=1. Every proper-factor term vanishes modulo m after that onset, leaving uv^n+v*u^n for arbitrary natural weights u,v. The existing nonunit-inclusive power-period theorem then applies to the two bases. No minimal onset or minimal period is claimed.

Evidence boundaries

Preregistration and the literature screen: https://github.com/the-omega-institute/trureturing/issues/24054. Literature completeness is ASSUMED-UNVERIFIED: rate-limited indices and unsearched full texts remain outside the inspected sources.

Independent finite checks are in the private experiments repository https://github.com/the-omega-institute/trureturing-experiments/tree/96efa433d49769b9ab2f8e0fb1263b9ca0725b6e/docs/reports/bala-a258899-a258903. The program check.py has SHA256 12c906c780a8b964892bf4dfe987eb5254ed0c2e089260d72b6999a1309a016a. Command: python3 docs/reports/bala-a258899-a258903/check.py; reported exit 0, with final line ALL_CHECKS_OK. These finite checks are independent evidence, not the universal proof.