{
  "abstract": "Following Erdős (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \\sum_{n\\leq x}h(ϕ(n)) \\asymp x\\left(\\frac{\\log\\log x}{\\log\\log\\log x}\\right)^{1/2}, $$ where $ϕ$ is the Euler totient function. The key ingredient is the Poisson random model for $ω(n,T)$, the number of the prime divisors of $n$ in a given subset of primes $T$, which was introduced in a recent work of Ford.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25341",
  "author_total": 2,
  "authors": [
    {
      "name": "Mikhail R. Gabdullin"
    },
    {
      "name": "Vitalii V. Iudelevich"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8g0kb0ayr7ye.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25341",
  "fulltext_url": null,
  "id": "c8g0kb0ayr7ye",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25341",
  "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": "W7214092774",
  "pdf_url": "https://arxiv.org/pdf/2609.25341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8g0kb0ayr7ye.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Distinct exponents in the prime factorization",
  "title_text": "Distinct exponents in the prime factorization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/c8g0kb0ayr7ye",
  "venue": "arXiv",
  "versions": []
}