Array read by upward antidiagonals

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers 2n+1 and 2n+3 for n > 0, at least one will be the product of 2 primes (not necessarily distinct).

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-105020,
  title        = {Array read by upward antidiagonals},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-105020}},
  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

A "Goldbach Conjecture" for this sequence: when there are $n$ terms between consecutive odd integers $2 n + 1$ and $2 n + 3$ for $n > 0$ , at least one will be the product of 2 primes (not necessarily distinct). Example: $n = 3$ for consecutive odd integers $a (7) = 7$ and $a (11) = 9$ and of the 3 sequence entries $a (8) = 12$ , $a (9) = 15$ and $a (10) = 16$ between them, one is the product of 2 primes $a (9) = 15 = 3 * 5$ . - _Michael Hiebl_, Jul 15 2007

Array read by upward antidiagonals: row $n$ ( $n \geq 0$ ) contains the numbers $m^{2} - n^{2}$ , $m \geq n + 1$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«105020».

theorem conjecture :
  ∀ (n i j : ℕ), 1 ≤ n →
    a i = 2 * n + 1 →
    a j = 2 * n + 3 →
    j = i + n + 1 →
    ∃ (k : ℕ),
      i < k ∧
      k < j ∧
      (a k).IsSemiprime

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

A105020

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.