{
  "abstract": "Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \\ge 2d(n!)$. Erdős, Graham, Ivić and Pomerance proved that, for infinitely many $n$, \\begin{equation*} K(n) > (1/9)(\\log n)(\\log_{2} n)(\\log_{4} n)/(\\log_{3} n)^3. \\end{equation*} 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15597",
  "author_total": 1,
  "authors": [
    {
      "name": "Tristan Freiberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45qnsnh7k7v8p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15597",
  "fulltext_url": null,
  "id": "45qnsnh7k7v8p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15597",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213266731",
  "pdf_url": "https://arxiv.org/pdf/2609.15597",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45qnsnh7k7v8p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15597",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15597"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Long runs of integers with small prime factors and the divisor function of $n!$",
  "title_text": "Long runs of integers with small prime factors and the divisor function of n!",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/45qnsnh7k7v8p",
  "venue": "arXiv",
  "versions": []
}