slug: oeis-a266489-diagonal-index-divisibility bibkey: hanna2016diagonalindex doi: null url: https://oeis.org/A266489 triage: theorem motivation_gids:

• D5/S1/Recurrence/Residue/DiagonalVanishingIndexDivisibility

A266489: clause C1, index divisibility on the frozen object

Problem

Quoted directly from the seat-local file oeis-A266489.src (seats/opB-diagidx2/oeis-A266489.src in the supplied workspace):

NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1. COMMENT: CONJECTURES: COMMENT: (C1) n divides a(n) for n>=1: A268293(n) = a(n)/n. COMMENT: (C2) a(n) == 0 (mod 2) for n>=2. AUTHOR: Paul D. Hanna, Feb 07 2016

The target is only clause (C1) for this entry. The formal object has a(0)=a(1)=1 and exponent e(n)=n. The formal vanishing condition is imposed for n>1, as in the quoted NAME.

Motivation

This is a first-tier OEIS conjecture attributed to Paul D. Hanna in 2016. The target is the unbounded assertion n divides this entry’s a(n) for every n>=1. This dossier and its claim account for this entry alone.

Gap

The supplied source labels the target a conjecture. This offline follow-up checks the entry-specific source and its connection to the delivered theorem; it performs no new external literature or revision-history search. Whether an earlier proof exists outside the supplied material remains unverified.

Route

The module constructs a normalized integer power series by a triangular coefficient update. Agreement of inverses preserves agreement of coefficients, and each update improves agreement by one degree. Stabilization produces the series; generating_equation and generating_unique establish its normalized vanishing equation and uniqueness.

The coefficient engine power_coefficient_identity comes from differentiating integer powers of a unit, including negative powers. In the triangular sum, put alpha=v_p(n). If v_p(m)<alpha, then v_p(n-m)=v_p(m), so the engine supplies the required prime-power factor in the inverse-power coefficient. If v_p(m)>=alpha, strong induction supplies it in a(e,m). Combining prime multiplicities, summing and negating proves index_power_divisibility. Its hypothesis n^k | e(n) for every natural n is sufficient; no claim is made that it is the weakest sufficient hypothesis.

For this entry specialize e(n)=n and k=1.

For this entry, agreement_a266489 identifies a (fun m => m) n with NegativePowerDiagonalModPrime.a 2 n at every index, using the frozen NegativePowerDiagonalModPrime.generating_unique. The named theorem states C1 about that frozen coefficient function itself. This transfer is the lane’s only genuine dependency edge to the frozen sibling module. The two modules have different exponent families; this lane does not generalize that module.

Falsifier

A natural index n>=1 at which n fails to divide this entry’s normalized coefficient would falsify the claimed conclusion. Here the coefficient in the conclusion is the frozen NegativePowerDiagonalModPrime.a 2 n. Finite agreement with DATA and finite divisibility checks cannot exclude an arbitrary later counterexample.

Evidence

• Lean module: D5/S1/Recurrence/Residue/DiagonalVanishingIndexDivisibility.lean.
• Entry-specific theorem and Scribe claim: hanna_conjecture_a266489.
• Exact conclusion for n : ℕ, hn : 1 ≤ n: (n : ℤ) ∣ NegativePowerDiagonalModPrime.a 2 n.
• General input: index_power_divisibility; coefficient engine: power_coefficient_identity; construction: generating_equation and generating_unique.
• Transfer to the existing object: agreement_a266489.

The orchestrator reports from results/verify-r28.py / results/verify-r28.out that this entry’s published DATA is reproduced exactly (15 terms), with zero violations of its divisibility conjecture for 1 <= n < 16. These are attributed finite supporting readings, not a proof or a fresh run by this seat.

Triage

theorem. This lane settles exactly A266489 (C1), n divides a(n) for n>=1. The separate (C2), the mod-2 congruence, was settled earlier by the frozen module NegativePowerDiagonalModPrime and is not this lane’s work. The claim is attached to hanna_conjecture_a266489, whose conclusion concerns NegativePowerDiagonalModPrime.a 2 via agreement_a266489; it does not claim C2.

ASSUMED-UNVERIFIED

The quotations and author date were read directly from this entry’s own local .src file; the file’s fidelity to the live OEIS page and revision history was not independently checked because this seat has no network. External OEIS-to-Lean identification and publication priority are not kernel-checked facts. No exhaustive literature search is claimed. The numerical readings above are supplied by the orchestrator and were not independently recomputed by this follow-up seat; they are support, not proof.