Ascending descending base exponent transform of 2^n

The first prime terms in this (always odd) sequence are a(1) = 3, a(3) = 41, and a(4) = 593. What is the next prime? The OEIS comment currently says a(5) = 543, but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457.

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Start working on it Submit a claim Follow

Cite

@misc{cairn-oeis-113271,
  title        = {Ascending descending base exponent transform of 2^n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-113271}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The first prime terms in this (always odd) sequence are $a (1) = 3$ , $a (3) = 41$ , and $a (4) = 593$ . What is the next prime?

The OEIS comment currently says $a(5) = 543$ , but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457. Consequently the search for the next prime must begin after index 4, not after index 5.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«113271». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture1 :
  answer(sorry) = a (sInf {n : ℕ | 4 < n ∧ (a n).Prime})

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

A113271

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.