Smallest index k > n such that (p_k+p_k+1)/(p_n+p_n+1) is an integer ≥ 2

Conjecture: the sequence is infinite, that is, for every n ≥ 1 there is some k > n with S(n) | S(k), so that a(n) is defined.

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-167918,
  title        = {Smallest index k > n such that (p_k+p_k+1)/(p_n+p_n+1) is an integer ≥ 2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-167918}},
  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

conjecture_infinite. Conjecture: the sequence is infinite, that is, for every $n \geq 1$ there is some $k > n$ with $S (n) ∣ S (k)$ , so that $a (n)$ is defined.

conjecture1. Conjecture: $f (n, k) = 2$ for infinitely many cases, where $k = a (n)$ .

No hypothesis on the existence of $a (n)$ is needed: if no suitable $k > n$ exists then $a (n) = 0$ and $S (0) = 4 < 2 S (n)$ , so such $n$ cannot be witnesses.

conjecture2. Open problem: is the ratio $f (n, k)$ bounded, where $k = a (n)$ ?

No hypothesis on the existence of $a (n)$ is needed: if no suitable $k > n$ exists then $a (n) = 0$ and $S (0) / S (n) = 4/ S (n) = 0$ for $n > 0$ , so such $n$ do not affect boundedness. When $a (n) \neq = 0$ we have $S (n) ∣ S (a (n))$ by definition, so the division is exact.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«167918» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture_infinite (n : ℕ) (hn : n > 0) : ∃ k > n, S n ∣ S k

theorem conjecture1 (M : ℕ) :
    ∃ n : ℕ, n ≥ M ∧ n > 0 ∧ S (a n) = 2 * S n

theorem conjecture2 : answer(sorry) ↔ ∃ C : ℕ, ∀ n : ℕ, n > 0 → S (a n) / S n ≤ C

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

A167918

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.