Number of refactorable numbers (A033950) $\le 10^n$

formal-conjectures · oeis · ams-11

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

Simon Colton conjectures that the number of refactorable numbers less than $x$ is at least
$\frac{x}{2\log x}$. This is an asymptotic claim, so we state it for sufficiently large $x$.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A111291
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/111291.lean
Local target: Corpus.OEIS111291.conjecture
Source SHA-256: eadbfe2c23d97305ae3dd6dea09fd219b2cc2a9c6cffb526341e38f0761e19e2
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record