Additive decompositions of π(T(n))

Conjecture (i): a(n) > 0 for all n > 4.

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-262403,
  title        = {Additive decompositions of π(T(n))},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-262403}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

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 (i): $a (n) > 0$ for all $n > 4$ .

conjecture_2. Conjecture (i): $a (n) = 1$ only for $n = 5, 6, 7, 10, 12, 32, 38, 445, 727$ .

conjecture_3. Conjecture (ii), first assertion: All numbers $π (T (n))$ ( $n = 1, 2, 3, \dots$ ) are pairwise distinct.

conjecture_4. Conjecture (ii), second assertion: If $\sum_{i = j}^{k} \frac{1}{π ( T ( i ))}$ and $\sum_{r = s}^{t} \frac{1}{π ( T ( r ))}$ with $1 < j \leq k$ and $j \leq s \leq t$ have the same fractional part but $(j, k) \neq = (s, t)$ , then $j = 2$ , $k = 5$ , and $s = t = 4$ .

Number of ways to write $π (T (n)) = π (T (k)) + π (T (m))$ with $1 < k < m < n$ , where $T (x) = \frac{x ( x + 1 )}{2}$ is the $x$ -th triangular number and $π (x)$ is the number of primes not exceeding $x$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«262403» (4 statements).

theorem conjecture_1 (n : ℕ) (hn : 4 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 4 < n) :
    a n = 1 ↔ n ∈ singletons

theorem conjecture_3 (m n : ℕ) (hm : 0 < m) (hn : 0 < n)
    (h : primeCountTriangular m = primeCountTriangular n) : m = n

theorem conjecture_4 (j k s t : ℕ) (hj : 1 < j) (hjk : j ≤ k) (hjs : j ≤ s)
    (hst : s ≤ t) (hne : (j, k) ≠ (s, t))
    (h : Int.fract (∑ i ∈ Finset.Icc j k, (primeCountTriangular i : ℚ)⁻¹) =
         Int.fract (∑ r ∈ Finset.Icc s t, (primeCountTriangular r : ℚ)⁻¹)) :
    j = 2 ∧ k = 5 ∧ s = 4 ∧ t = 4

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

A262403
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.