Smallest deficient multiplier for non-deficiency

Conjecture: For every n ≥ 2, a(n) is 1, 3, or a power of 2.

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-215926,
  title        = {Smallest deficient multiplier for non-deficiency},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-215926}},
  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: For every $n \geq 2$ , $a (n)$ is $1$ , $3$ , or a power of $2$ .

conjecture_2. Conjecture: The first occurrence of $2^{m}$ happens at A014210 $(m)$ .

We require $m \geq 1$ because for $m = 0$ , the first occurrence of $2^{0} = 1$ is at $n = 6$ , whereas $A014210 (0) = 2$ gives $a (2) = 3$ .

For $n \geq 2$ , $a (n)$ is the smallest deficient number $k$ such that the product $k \cdot n$ is non-deficient (i.e., perfect or abundant).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«215926» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : 2 ≤ n) :
    a n = 1 ∨ a n = 3 ∨ (a n).isPowerOfTwo

theorem conjecture_2 (m : ℕ) (hm : 1 ≤ m) :
    IsLeast {n : ℕ | 2 ≤ n ∧ a n = 2 ^ m} (nextPrimeTwoPow m)

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

A215926

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.