Factorial distance to nearest square

formal-conjectures · oeis · ams-11

The sequence is defined as
$$a(n) =
\mathrm{round}\left(\frac{\mathrm{round}(\sqrt{n!})}{\left|(\mathrm{round}(\sqrt{n!}))^2 -
n!\right|}\right)$$
for $n \ge 2$.

This sequence suggests that the distance between a factorial and the closest power is
tightly bounded.

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/A145355
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/145355.lean
Local target: Corpus.OEIS145355.conjecture
Source SHA-256: 3689ce06e591dc66cd9580c41809587f875bed95783663b7cc10c10592f7bb4f
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