$a(n)$ = the number of values of $k <= 10^n$ such that $\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime

formal-conjectures · oeis · ams-11

Since $\sqrt{k(k+1)(k+2)(k+3)+1} = k^2 + 3k + 1$,
$a(n) = \#\{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} \}.$

Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$
where A006880(n) is the number of primes $\le 10^n$.

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/A115366
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/115366.lean
Local target: Corpus.OEIS115366.conjecture
Source SHA-256: ff2cdca8c02536491fe413ad9dd7812f41500d4b8a64754b6d943ee66206dd1a
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