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, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.

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BibTeX RIS

BibTeX

@misc{nichols2026surjectivity,
  title = {{Surjectivity of the Enots Wolley Sequence}},
  author = {Nathan Myles Nichols},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18054},
  eprint = {2609.18054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18054},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Surjectivity of the Enots Wolley Sequence
AU  - Nathan Myles Nichols
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.18054
UR  - https://arxiv.org/abs/2609.18054
AB  - We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.
ER  -

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