Number of integers 0 < k < n such that k φ(n - k) + 1 is a square

Conjecture (i): a(n) > 0 if n is not a divisor of 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-234246,
  title        = {Number of integers 0 < k < n such that k φ(n - k) + 1 is a square},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-234246}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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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): $a (n) > 0$ if $n$ is not a divisor of $6$ .

conjecture_2. Conjecture (i): The only values of $n$ with $a (n) = 1$ are $4, 5, 8, 9, 12, 13, 24, 33, 49$ .

conjecture_3. Conjecture (ii): If $n \geq 60$ , then $k + ϕ (n - k)$ is a square for some $0 < k < n$ .

conjecture_4. Conjecture (ii): If $n > 60$ , then $σ (k) + ϕ (n - k)$ is a square for some $0 < k < n$ , where $σ (k)$ is the sum of all positive divisors of $k$ .

conjecture_5. Conjecture (iii): If $n > 7$ is not equal to $10$ or $20$ , then $ϕ (k) ϕ (n - k) + 1$ is a square for some $0 < k < n$ .

conjecture_6. Conjecture (iv): If $n > 7$ is not equal to $10$ or $19$ , then $(ϕ (k) + ϕ (n - k)) /2$ is a triangular number for some $0 < k < n$ .

Let $ϕ$ denote Euler's totient function. The sequence $a (n)$ counts the integers $0 < k < n$ such that $k ϕ (n - k) + 1$ is a square.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«234246» (6 statements).

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

theorem conjecture_2 (n : ℕ) :
    a n = 1 ↔ n ∈ ({4, 5, 8, 9, 12, 13, 24, 33, 49} : Finset ℕ)

theorem conjecture_3 (n : ℕ) (hn : 60 ≤ n) :
    ∃ k, 0 < k ∧ k < n ∧ IsSquare (k + (n - k).totient)

theorem conjecture_4 (n : ℕ) (hn : 60 < n) :
    ∃ k, 0 < k ∧ k < n ∧ IsSquare (σ 1 k + (n - k).totient)

theorem conjecture_5 (n : ℕ) (hn : 7 < n) (hne10 : n ≠ 10) (hne20 : n ≠ 20) :
    ∃ k, 0 < k ∧ k < n ∧ IsSquare (k.totient * (n - k).totient + 1)

theorem conjecture_6 (n : ℕ) (hn : 7 < n) (hne10 : n ≠ 10) (hne19 : n ≠ 19) :
    ∃ k, 0 < k ∧ k < n ∧ ∃ m : ℕ, k.totient + (n - k).totient = m * (m + 1)

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

A234246

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.