Number of integers 0 < k < n such that (k + 1)^φ(n - k) + k 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-234360,
  title        = {Number of integers 0 < k < n such that (k + 1)^φ(n - k) + k is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-234360}},
  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_3. Conjecture (ii): If $n > 1$ , then $k (k + 1)^{ϕ (n - k)} + 1$ is prime for some $0 < k < n$ .

conjecture_4. Conjecture (ii): If $n > 3$ , then $k (k + 1)^{ϕ (n - k) /2} - 1$ is prime for some $0 < k < n$ .

For $n \geq 1$ , $a (n)$ counts the integers $0 < k < n$ for which $(k + 1)^{ϕ (n - k)} + k$ is prime, where $ϕ$ is Euler's totient function.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«234360» (3 statements).

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

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

theorem conjecture_4 (n : ℕ) (hn : 3 < n) :
    ∃ k, 0 < k ∧ k < n ∧
      ∃ p : ℕ, p.Prime ∧ (p + 1) ^ 2 = k ^ 2 * (k + 1) ^ (n - k).totient

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

A234360
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

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.