Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\ge 2$

formal-conjectures · oeis · ams-11

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

We assume $a(n) \ne 0$ (i.e., that a suitable $k > n$ always exists), as sInf evaluates to $0$
on an empty set.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A167918
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/167918.lean
Local target: Corpus.OEIS167918.conjecture1
Source SHA-256: e6cf437ccac7561b5466a14c42cd6f4dade71de51132cb0f9539401cfb6e3793
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record