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Level A · Machine-checkable Hard Number theory P-oeis-103425

a(n) = 3a(n-1) + a(n-2) - 3a(n-3)

The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free?

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-103425,
  title        = {a(n) = 3a(n-1) + a(n-2) - 3a(n-3)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-103425}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

The current sequence contains primes, including . Is there an weighted tribonacci sequence with relatively prime which is prime-free?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«103425». 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) ↔
    ∃ (a b c : ℤ) (x : ℕ → ℤ),
      Nat.gcd (Int.gcd a b) c.natAbs = 1 ∧
      IsWeightedTribonacci a b c x ∧
      ∀ n, ¬ (x n).natAbs.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

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.