Number of prime powers strictly between n-th prime and (n+1)-th prime

It is conjectured that a(n) ≤ 2 for all n.

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.

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Cite

@misc{cairn-oeis-80101,
  title        = {Number of prime powers strictly between n-th prime and (n+1)-th prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-80101}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

It is conjectured that $a (n) \leq 2$ for all $n$ .

The sequence $a (n)$ is the number of prime powers $k$ strictly between the $n$ -th prime $p_{n}$ and the $(n + 1)$ -th prime $p_{n + 1}$ : $p_{n} < k < p_{n + 1}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«80101».

theorem conjecture (n : ℕ) : a n ≤ 2

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

A080101

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.