Number of refactorable numbers (A033950) ≤ 10^n

Colton's conjecture [Co99] as stated by Zelinsky [Ze02]: for every n, the number of refactorable numbers ≤ n is at least half the number of primes ≤ n, i.e. π(n) ≤ 2 T(n).

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-111291,
  title        = {Number of refactorable numbers (A033950) ≤ 10^n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-111291}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Colton's conjecture [Co99] as stated by Zelinsky [Ze02]: for every $n$ , the number of refactorable numbers $\leq n$ is at least half the number of primes $\leq n$ , i.e. $π (n) \leq 2 T (n)$ .

Zelinsky [Ze02] proved this for all sufficiently large $n$ (see conjecture), with an explicit bound of $7.42 \cdot 1 0^{13}$ beyond which it holds; the remaining range is open.

A number $k$ is refactorable if its number of divisors, $τ (k)$ , divides $k$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«111291».

theorem colton_conjecture : ∀ n : ℕ, Nat.primeCounting n ≤ 2 * countRefactorableNat n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

A111291
[Co99] Colton, S., Refactorable numbers - a machine invention. J. Integer Seq. 2 (1999), Article 99.1.2.
[Ze02] Zelinsky, J., Tau numbers: a partial proof of a conjecture and other results. J. Integer Seq. 5 (2002), Article 02.2.8.
[Sp85] Spiro, C., How often is the number of divisors of n a divisor of n? J. Number Theory 21 (1985), 81--100.

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.