Odd part of the digital sum of 3^n divided by the maximal power of 3

Conjecture: Consider the sequence b(n) such that b(1)=2, b(2)=3, and for n ≥ 3, b(n)=a(n-2). If we apply the Eratosthenes-like sieve to b(n) and remove 1s, then we obtain a sequence of primes. - _Vladimir Shevelev_, Dec 12 2014

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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-oeis-250131,
  title        = {Odd part of the digital sum of 3^n divided by the maximal power of 3},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-250131}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Foeis-250131.json)](https://cairn-commons.com/problems/oeis-250131)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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. Conjecture: Consider the sequence $b (n)$ such that $b (1) = 2$ , $b (2) = 3$ , and for $n \geq 3$ , $b (n) = a (n - 2)$ . If we apply the Eratosthenes-like sieve to $b (n)$ and remove $1$ s, then we obtain a sequence of primes.

_Vladimir Shevelev_, Dec 12 2014

conjecture_2. Stronger variant: the sieved sequence $b (n)$ yields the set of all primes (in some perturbed order).

Note: This stronger statement is not explicitly written on the A250131 OEIS page (which only states conjecture_1 and notes via _Peter J. C. Moses_ that "these primes follow with some perturbation of order"), but is implied by prepending $b (1) = 2, b (2) = 3$ (which otherwise have no effect on the sieve since $g cd (a (n), 6) = 1$ ) and by _Vladimir Shevelev_'s analogous conjecture in the cross-referenced entries A221858 and A225039.

The sequence $a (n)$ is the odd part of the base- $10$ digital sum of $3^{n}$ divided by the maximal possible power of $3$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«250131» (2 statements).

theorem conjecture_1 : ∀ n ∈ sievedIndices, (b n).Prime

theorem conjecture_2 : b '' sievedIndices = {p : ℕ | p.Prime}

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

A250131

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.