Representations of n as x^2 + φ(y^2) + φ(z^2)

Conjecture (i): a(n) > 0 for all n > 6.

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-262781,
  title        = {Representations of n as x^2 + φ(y^2) + φ(z^2)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-262781}},
  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 $n > 6$ .

conjecture_2. Conjecture (i): $a (n) = 1$ only for $n = 3, 5, 9, 10, 17, 20, 24, 25, 31, 36, 45, 73, 80, 101, 136, 145, 388, 649$ .

conjecture_3. Conjecture (ii): For any integer $n > 4$ , we can write $2 n$ as $ϕ (p^{2}) + ϕ (x^{2}) + ϕ (y^{2})$ with $p$ prime and $p \leq x \leq y$ .

Number of ordered ways to write $n$ as $x^{2} + ϕ (y^{2}) + ϕ (z^{2})$ ( $x \geq 0$ and $0 < y \leq z$ ) with $y$ or $z$ prime, where $ϕ (\cdot)$ is Euler's totient function.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«262781» (3 statements).

theorem conjecture_1 (n : ℕ) (hn : 6 < n) : 0 < a n

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

theorem conjecture_3 (n : ℕ) (hn : 4 < n) :
    ∃ p x y : ℕ, p.Prime ∧ p ≤ x ∧ x ≤ y ∧
      2 * n = Nat.totient (p ^ 2) + Nat.totient (x ^ 2) + Nat.totient (y ^ 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

A262781

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.