Catalan numbers modulo 6

It is conjectured that the only k which yield a(2^k - 1) = 1 are k = 0, 1 and 5.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-259667,
  title        = {Catalan numbers modulo 6},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-259667}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. It is conjectured that the only $k$ which yield $a (2^{k} - 1) = 1$ are $k = 0, 1$ and $5$ .

conjecture_2. Conjecture: The only $k$ which yield $a (2^{k} - 1) = 5$ are $k = 2, 8$ .

The sequence $a (n) = C_{n} mod 6$ , where $C_{n} = \frac{1}{n + 1} (n 2 n)$ is the $n$ -th Catalan number (A000108).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«259667» (2 statements).

theorem conjecture_1 (k : ℕ) :
    a (2 ^ k - 1) = 1 ↔ k = 0 ∨ k = 1 ∨ k = 5

theorem conjecture_2 (k : ℕ) :
    a (2 ^ k - 1) = 5 ↔ k = 2 ∨ k = 8

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A259667

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.