a(0) = 1, a(n) = a(n-1)a(n-1) + 2

Prime for a(1) = 3, a(2) = 11, a(4) = 15131; semiprime for a(3) = 123 = 3 41, a(5) = 228947163 = 3 76315721. a(6), added by Jonathan Vos Post, has 4 prime factors. a(7) = 41 811^2 106693969 317171188688357726699 8272236925540996054440172449761. When is the next prime in the sequence?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-oeis-102847,
  title        = {a(0) = 1, a(n) = a(n-1)a(n-1) + 2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-102847}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Prime for $a (1) = 3$ , $a (2) = 11$ , $a (4) = 15131$ ; semiprime for $a (3) = 123 = 3 * 41$ , $a (5) = 228947163 = 3 * 76315721$ . $a (6)$ , added by Jonathan Vos Post, has 4 prime factors. $a (7) = 41 * 81 1^{2} * 106693969 * 317171188688357726699 * 8272236925540996054440172449761$ . When is the next prime in the sequence?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«102847». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture : answer(sorry) = sInf {n : ℕ | 4 < n ∧ (a n).Prime}

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

A102847

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.