Number of primes $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$

formal-conjectures · oeis · ams-11

The sequence $a(n)$ counts the number of primes $p$ in the interval $[2^n, 2^n + p_n]$,
where $p_n$ is the $n$-th prime ($p_1 = 2, p_2 = 3, \dots$):
$$a(n) = |\{p \text{ prime} \mid 2^n \le p \le 2^n + p_n\}|$$
for $n \ge 1$, and $a(0) = 0$.

For any $n > 0$, is there always at least one prime $p$ such that
$2^n \le p \le 2^n + \mathrm{prime}(n)$?
(checked up to $n = 250$). In this case, that would be stronger than the Schinzel conjecture:
"for $m > 1$ there's at least one prime $p$ such that $m \le p \le m + \log(m)^2$" since,
for $n > 2$, $\mathrm{prime}(n) < \log(2^n)^2 = n^2 \log(2)$.

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/A069923
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/69923.lean
Local target: Corpus.OEIS69923.conjecture
Source SHA-256: 96cb96097c686ec70883f241119076b77a0587c9c96e0840931a6b44473ab6d2
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.

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