{
  "abstract": "The paper defines a certain function \\(χ(n)\\) on the set of odd integers \\(n \\ge 3\\). The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if \\(n\\) is a prime number or an odd Giuga number. Thus, the integrality of \\(χ(n)\\) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt--Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers \\(p\\) for which \\(χ(p)=p-1\\) and Novák--Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes \\(P_μ\\) of primes with a fixed value \\(χ(p)=μ\\), a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number \\(B_{p-1}\\). For some \\(μ\\), explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18607",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrei K. Svinin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3qza7tnaq42v4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18607",
  "fulltext_url": null,
  "id": "3qza7tnaq42v4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18607",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7170046703",
  "pdf_url": "https://arxiv.org/pdf/2607.18607",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3qza7tnaq42v4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2607.18607",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18607"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a certain arithmetic function defined via Bernoulli numbers",
  "title_text": "On a certain arithmetic function defined via Bernoulli numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3qza7tnaq42v4",
  "venue": "arXiv",
  "versions": []
}