Divisibility of 2^n + 1 by n

All members of the sequence A56777 come from prime quadruples.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-56777,
  title        = {Divisibility of 2^n + 1 by n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-56777}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

All members of the sequence A56777 come from prime quadruples.

A56777 lists composite numbers $n$ satisfying both $φ (n + 12) = φ (n) + 12$ and $σ (n + 12) = σ (n) + 12$ .

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«56777».

theorem comesFromPrimeQuadruple_of_a {n : ℕ} (h : A n) : ComesFromPrimeQuadruple n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A56777

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.