A proof of Colton's conjecture on refactorable numbers

A positive integer is refactorable (a tau number) if it is divisible by its number of divisors. Let T(x) count the refactorable numbers up to x and π(x) the primes up to x. In 1999 S. Colton conjectured that π(n) ≤ 2T(n) for every n; in 2002 J. Zelinsky proved this for n > 7.42·1013. This note proves it for every positive integer n, with equality only at n = 7. The proof combines an exact computation for n ≤ 1010, a lower bound for T(x) built from Zelinsky's generators and evaluated with Dusart's explicit bounds for π(x) on [1010, 2.07·1013], and a short analytic argument beyond 2·1013. Every computation was repeated by an independent implementation. We also find T(1012) = 26,568,322,474. The record contains the note (PDF), the programs, their outputs, the list of generators used, and an independent review. The code is also on GitHub. This work was done with the assistance of a large language model; the note describes its role.