{
  "abstract": "For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\\le m\\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\\le(2\\log2+o(1))\\,n/\\log n$. This improves the bounds of order $n/(\\log n)^{1/2-\\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\\log2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10902",
  "author_total": 1,
  "authors": [
    {
      "name": "Scott D. Hughes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8n6dhbk2hzx3t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10902",
  "fulltext_url": null,
  "id": "8n6dhbk2hzx3t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10902",
  "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": "W7212284489",
  "pdf_url": "https://arxiv.org/pdf/2609.10902",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8n6dhbk2hzx3t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10902",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10902"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sums of distinct divisors of factorials",
  "title_text": "Sums of distinct divisors of factorials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/8n6dhbk2hzx3t",
  "venue": "arXiv",
  "versions": []
}