Representations n = k^2 + m (0 < k^2 ≤ m) with σ(k^2) + φ(m) prime

Conjecture (i): a(n) > 0 for all n > 1. - _Zhi-Wei Sun_

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-233544,
  title        = {Representations n = k^2 + m (0 < k^2 ≤ m) with σ(k^2) + φ(m) prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-233544}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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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 $n > 1$ .

_Zhi-Wei Sun_

conjecture_2. Conjecture (ii): Any integer $n > 1$ can be written as $k + m$ with $k > 0$ and $m > 0$ such that $σ (k)^{2} + ϕ (m)$ is prime.

_Zhi-Wei Sun_

conjecture_3. Conjecture (ii): Any integer $n > 1$ can be written as $k + m$ with $k > 0$ and $m > 0$ such that $σ (k) + ϕ (m)^{2}$ is prime.

_Zhi-Wei Sun_

Let $σ$ denote the sum-of-divisors function and $ϕ$ denote Euler's totient function. The sequence $a (n)$ counts the representations $n = k^{2} + m$ ( $0 < k^{2} \leq m$ ) such that $σ (k^{2}) + ϕ (m)$ is prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«233544» (3 statements).

theorem conjecture_1 (n : ℕ) (hn : 1 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 1 < n) :
    ∃ k m : ℕ, 0 < k ∧ 0 < m ∧ n = k + m ∧ Nat.Prime (σ 1 k ^ 2 + m.totient)

theorem conjecture_3 (n : ℕ) (hn : 1 < n) :
    ∃ k m : ℕ, 0 < k ∧ 0 < m ∧ n = k + m ∧ Nat.Prime (σ 1 k + m.totient ^ 2)

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

A233544

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.