Number of positive integers k < n such that n^2 - π(k n) is prime

Conjecture (i): a(n) > 0 for all n > 1.

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-238568,
  title        = {Number of positive integers k < n such that n^2 - π(k n) is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-238568}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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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. Conjecture (i): $a (n) > 0$ for all $n > 1$ .

conjecture_2. Conjecture (i): $a (n) = 1$ only for $n \in {2, 3, 4, 8, 10, 24, 41}$ .

conjecture_3. Conjecture (ii): For any integer $n > 6$ , there is a positive integer $k < n$ with $n^{2} + π (k n) - 1$ prime.

conjecture_4. Conjecture (iii), part 1: If $n > 2$ , then $π (n^{2}) - π (k n)$ is prime for some $0 < k < n$ .

conjecture_5. Conjecture (iii), part 2: If $n > 1$ , then $π (n^{2}) + π (k n) - 1$ is prime for some $0 < k < n$ .

The sequence $a (n)$ counts the number of positive integers $k < n$ such that $n^{2} - π (k n)$ is prime, where $π (x)$ is the prime counting function: $a (n) = ∣ {0 < k < n : n^{2} - π (k n) is prime} ∣$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«238568» (5 statements).

theorem conjecture_1 (n : ℕ) (hn : 1 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) :
    a n = 1 ↔ n ∈ ({2, 3, 4, 8, 10, 24, 41} : Set ℕ)

theorem conjecture_3 (n : ℕ) (hn : 6 < n) :
    ∃ k : ℕ, 0 < k ∧ k < n ∧ Nat.Prime (n ^ 2 + Nat.primeCounting (k * n) - 1)

theorem conjecture_4 (n : ℕ) (hn : 2 < n) :
    ∃ k : ℕ, 0 < k ∧ k < n ∧
      Nat.Prime (Nat.primeCounting (n ^ 2) - Nat.primeCounting (k * n))

theorem conjecture_5 (n : ℕ) (hn : 1 < n) :
    ∃ k : ℕ, 0 < k ∧ k < n ∧
      Nat.Prime (Nat.primeCounting (n ^ 2) + Nat.primeCounting (k * n) - 1)

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

A238568
Problems on combinatorial properties of primes by Zhi-Wei Sun (2014)

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.