{
  "abstract": "Let $p_n$ be the $n$th prime, and let $r,s\\ge2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a s^b}$ with integers $a,b\\ge0$. Our asymptotic formula for this count has relative error $o(1)$ and is explicit down to the multiplicative constant. For $(r,s)=(2,3)$ these integers are the prime codes of the ordinals below $ω^{ω^2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19434",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehdi Golafshan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ep87ppfvyas1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19434",
  "fulltext_url": null,
  "id": "2ep87ppfvyas1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19434",
  "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": "W7213610992",
  "pdf_url": "https://arxiv.org/pdf/2609.19434",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ep87ppfvyas1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19434",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19434"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Asymptotic counting of integers with prime factors $p_{r^a s^b}$",
  "title_text": "Asymptotic counting of integers with prime factors p_r^a s^b",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/2ep87ppfvyas1",
  "venue": "arXiv",
  "versions": []
}