Number of 0 < k < n/2 such that φ(k)φ(n-k) is a square

Conjecture: (i) a(n) > 0 for all n > 8.

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-236998,
  title        = {Number of 0 < k < n/2 such that φ(k)φ(n-k) is a square},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-236998}},
  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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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 $n > 8$ .

conjecture_2. Conjecture: (ii) If $n > 20$ , then $ϕ (k) ϕ (n - k) + 1$ is a square for some $0 < k < n /2$ .

conjecture_3. Conjecture: (iii) If $n > 1$ is not among $4, 7, 60, 199, 267$ , then $k ϕ (n - k)$ is a square for some $0 < k < n$ .

Let $ϕ$ denote Euler's totient function. The sequence $a (n)$ counts the number of integers $0 < k < n /2$ such that $ϕ (k) ϕ (n - k)$ is a square.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«236998» (3 statements).

theorem conjecture_1 (n : ℕ) (hn : 8 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 20 < n) :
    ∃ k : ℕ, 0 < k ∧ 2 * k < n ∧ IsSquare (k.totient * (n - k).totient + 1)

theorem conjecture_3 (n : ℕ) (hn : 1 < n) (h4 : n ≠ 4) (h7 : n ≠ 7) (h60 : n ≠ 60)
    (h199 : n ≠ 199) (h267 : n ≠ 267) :
    ∃ k : ℕ, 0 < k ∧ k < n ∧ IsSquare (k * (n - k).totient)

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

A236998

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.