Surjectivity of the Enots Wolley Sequence

Abstract

We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose an eligible integer is omitted, and let T be its finite set of prime divisors. There is a finite cutoff such that any maximal run of terms divisible by at least one prime of T and beginning after the cutoff starts with a term divisible by every prime of T and has length at most two. If infinitely many such runs occur, then terms divisible by some but not all primes of T can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite inequality at arbitrarily large scales: after discarding a negligible exceptional set, a weighted double count produces a fixed-factor excess of the former terms. Hence only finitely many such runs occur. Prime recurrence then forces every sufficiently late term to be divisible by some prime of T. Finally, a disjoint-cover argument rules out any finite eventual prime cover, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.