Long runs of integers with small prime factors and the divisor function of n!
Abstract
Let d be the divisor function, and let K(n) be the least positive integer K for which d((n + K)!) ≥ 2d(n!). Erdős, Graham, Ivić and Pomerance proved that, for infinitely many n, K(n) > (1/9)(log n)(log2 n)(log4 n)/(log3 n)3. We improve upon this by a factor of order log3 n, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of K(n): what is needed is a weighted variant of the Erdős--Rankin construction. We follow the method of Erdős, Graham, Ivić and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.
Related
- Asymptotic counting of integers with prime factors pra sb
- Note on the largest prime factors of consecutive integers
- Triple divisor type functions in arithmetic progressions to prime power moduli
- A Joint Local Limit Theorem for Prime Divisor Counting Functions
- A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor
BibTeX
@misc{freiberg2026long,
title = {{Long runs of integers with small prime factors and the divisor function of $n!$}},
author = {Tristan Freiberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15597},
eprint = {2609.15597},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15597},
}
RIS
TY - UNPB
TI - Long runs of integers with small prime factors and the divisor function of n!
AU - Tristan Freiberg
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15597
UR - https://arxiv.org/abs/2609.15597
AB - Let d be the divisor function, and let K(n) be the least positive integer K for which d((n + K)!) ≥ 2d(n!). Erdős, Graham, Ivić and Pomerance proved that, for infinitely many n, K(n) > (1/9)(log n)(log_2 n)(log_4 n)/(log_3 n)^3. We improve upon this by a factor of order log_3 n, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of K(n): what is needed is a weighted variant of the Erdős--Rankin construction. We follow the method of Erdős, Graham, Ivić and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.
ER -