slug: oeis-a396793-iterate-product-nine-mod-three bibkey: hanna2026a396793 doi: null url: https://oeis.org/A396793 triage: theorem motivation_gids:

• D5/S1/Recurrence/Residue/IterateProductNineModThree

Divisibility of the A396793 coefficients

Problem

OEIS A396793, Paul D. Hanna, Jun 07 2026, states:

NAME:

G.f. A(x) satisfies A(x) * A(A(x)) = x^2 + 9*x^3.

COMMENT:

Conjecture: a(n) == 0 (mod 3) for n > 1.

These verbatim quotations are recorded in Library/Recurrence/hanna2026a396793.md. Here A is an integer formal power series normalized by A(0) = 0 and a(1) = 1, and a(n) is its coefficient of x^n.

Motivation

This is a first-tier OEIS conjecture from the 2026 entry. The target proves the assertion for every index n > 1. KPI: open problems resolved.

Gap

The search seat reported no proof found after reading the OEIS entry and revision history on 2026-09-09 and searching the identifier on arXiv, MathOverflow, and GitHub. This Stage-B seat has no network and did not independently repeat those searches. Absence of a proof in that search scope does not establish first-publication priority.

Route

The escape content is lift_mod_nine: if f(0) = 0 and f is coefficientwise congruent to X modulo 3, then f * iterate f 2 is coefficientwise congruent to X^2 modulo 9. Writing f = X + 3B gives f composed with f congruent to X + 6B, and f times that composition congruent to X^2 modulo 9.

Together with the proved coefficient multiplier 3 (product_top), this lift makes the degree-by-degree corrections exact integers divisible by 3. Their stabilized limit proves existence; generatingSeries selects that existential witness. generating_equation proves A * iterate A 2 = X^2 + 9X^3, A(0) = 0, and a(1) = 1. The NAME’s A(A(x)) is the frozen compositional iterate at 2 from D5/S1/Recurrence/Invariants/CompositionalIterateCongruence. generating_unique covers every normalized integer solution.

hanna_conjecture proves n > 1 implies 3 divides a(n), by applying the lift to truncated prefixes and comparing them with the solution using the coefficient multiplier. The reduced series over ZMod 3 is X, selected by the lift rather than by a characteristic-3 uniqueness claim. The target has generality I because it imports the frozen D5/S1/Recurrence/Invariants/CompositionalIterateCongruence, itself generality G.

Falsifier

An index n > 1 for the normalized solution with 3 not dividing a(n) would contradict the assertion. The orchestrator’s exact numerical check is supporting evidence only; the theorem has no finite index bound.

Evidence

• Lean module: D5/S1/Recurrence/Residue/IterateProductNineModThree.lean.
• Main theorem: hanna_conjecture.
• Companion theorems: lift_mod_nine, generating_equation, generating_unique.
• Axioms: std3 (propext, Classical.choice, Quot.sound), as reported by the implementation seat for each of the four public theorems.

Triage

theorem. The formal proof closes the universal coefficient assertion for the unique normalized integer solution of the quoted generating equation.

ASSUMED-UNVERIFIED

The quotations were supplied by the orchestrator and copied from the Library note. The OEIS revision history was read by the search seat, not by this seat. The reported literature scope was the OEIS entry and revision history on 2026-09-09 and identifier searches on arXiv, MathOverflow, and GitHub; this seat had no network. No exhaustive literature search or first-publication priority is claimed. Source-to-Lean identification is not a kernel-checked fact.