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

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Statement

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

Zelinsky [Ze02] proved this for all sufficiently large nn (see conjecture), with an explicit bound of 7.42⋅10137.42 \cdot 10^{13} beyond which it holds; the remaining range is open.

Lean statement

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

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/OEIS/111291.lean.

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