{
  "abstract": "For each fixed positive integer $h$, we study the divisibility relation $σ(n)\\midσ(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\\log x)^2)$. We also study the proportionality equation $σ(n+h)=λσ(n)$. For every fixed nonzero integer $h$, uniformly for all real $λ>0$, the number of solutions up to $x$ is $O(x/\\sqrt{\\log\\log\\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $σ(n+1)=2σ(n)$; the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $σ(n+h)=kσ(n)$ has infinitely many positive integer solutions for every fixed $h,k\\ge1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04980",
  "author_total": 2,
  "authors": [
    {
      "name": "Amirali Fatehizadeh"
    },
    {
      "name": "Florian Luca"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ebe0xf3zznzj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04980",
  "fulltext_url": null,
  "id": "9ebe0xf3zznzj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04980",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7211909035",
  "pdf_url": "https://arxiv.org/pdf/2609.04980",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ebe0xf3zznzj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04980",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04980"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Divisibility Relation $σ(n)\\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture",
  "title_text": "On the Divisibility Relation σ(n)|σ(n+h) and a Generalized Erdős--Sierpiński Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/9ebe0xf3zznzj",
  "venue": "arXiv",
  "versions": []
}