Additive decompositions of π(n(n+1)/2 + 1)

Conjecture: a(n) > 0 for all n > 3.

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.

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Cite

@misc{cairn-oeis-262446,
  title        = {Additive decompositions of π(n(n+1)/2 + 1)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-262446}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. Conjecture: $a (n) > 0$ for all $n > 3$ .

conjecture_2. Conjecture: $a (n) = 1$ only for $n = 4, 6, 11, 21, 54, 253, 325$ .

Number of ways to write $\pi(\frac{n(n+1)}{2} + 1) = \pi(\frac{k(k+1)}{2} + 1) + \pi(\frac{m(m+1)}{2} + 1)$ with $0 < k < m < n$ , where $π (x)$ is the prime-counting function.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«262446» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : 3 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 3 < n) :
    a n = 1 ↔ n ∈ singletons

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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

A262446
Problems on combinatorial properties of primes by Zhi-Wei Sun, arXiv:1402.6641 (2014)

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.