Divisibility of $2^n + 1$ by $n$

formal-conjectures · oeis · ams-11

A56777 lists composite numbers $n$ satisfying both $\varphi(n+12) = \varphi(n) + 12$ and
$\sigma(n+12) = \sigma(n) + 12$.

The conjectures state identities connecting A56777 and prime quadruples (A7530), as
well as congruences satisfied by the members of A56777.

All members of the sequence A56777 come from prime quadruples.

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/A56777
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/56777.lean
Local target: Corpus.OEIS56777.comesFromPrimeQuadruple_of_a
Source SHA-256: 99426cebc7f8853b4296d7920415205e6d85927208a9760b302a1e7b35bad928
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