{
  "abstract": "For a positive integer $t$, let $d_t$ denote the natural density of the set of $n$ for which $t$ is a sum of distinct divisors of $n$. Erdős proved that $d_t$ exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether $d_t \\sim c_3/(\\log t)^{c_4}$. We record the explicit bounds \\[ ν_t \\le d_t \\ll \\frac{(\\log\\log t)^{δ-3/2}}{(\\log t)^δ}, \\qquad ν_t = \\frac{K}{\\log t}\\Bigl(1 + O\\Bigl(\\frac{\\log\\log t}{\\log t}\\Bigr)\\Bigr), \\] where $δ= 0.086071\\ldots$ is the Erdős--Ford--Tenenbaum constant, $K=c\\,e^{-γ}$, and $c=1.33607\\ldots$ is the practical-number constant. Consequently, if Erdős's asymptotic holds, then $δ<c_4\\le 1$. A two-prime construction, using a half-scale sumset to obtain full residue coverage, then yields the pointwise excess \\[ \\liminf_{t\\to\\infty}(\\log t)\\,(d_t-ν_t) \\ge KI, \\] where $I=\\int_0^2 G(w)\\,\\mathrm{d}w=0.05887\\ldots$ is an explicit elementary integral. In particular $d_t\\ge 0.79/\\log t$ for every sufficiently large $t$, and $d_t$ is not asymptotic to $K/\\log t$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25446",
  "author_total": 1,
  "authors": [
    {
      "name": "Scott D. Hughes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4x28ex96dhab.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25446",
  "fulltext_url": null,
  "id": "f4x28ex96dhab",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25446",
  "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": "W7214118966",
  "pdf_url": "https://arxiv.org/pdf/2609.25446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4x28ex96dhab.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The density of sums of distinct divisors",
  "title_text": "The density of sums of distinct divisors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/f4x28ex96dhab",
  "venue": "arXiv",
  "versions": []
}