{
  "abstract": "Andrews and Newman introduced the minimal excludant $\\mathrm{mex}(λ)$ of an integer partition $λ$ and studied the summatory function $σ\\mathrm{mex}(n)$, and Baruah et al. refined this to the odd- and even-restricted functions $σ_o\\mathrm{mex}(n)$ and $σ_e\\mathrm{mex}(n)$. In this paper, we introduce and study the overpartition analogue $\\overline{σ_o\\mathrm{mex}}(n)$, defined as the sum of odd minimal excludants over all overpartitions of $n$. We first derive the exact generating function for $\\overline{σ_o\\mathrm{mex}}(n)$, and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary $q$-series arguments, together with a weight-one eta-quotient identity for $φ(q)^2$ verified via the Gordon--Hughes--Newman--Ligozat criterion, we establish an infinite family of congruences satisfied by $\\overline{σ_o\\mathrm{mex}}(n)$. Consequently, we obtain that $\\overline{σ_o\\mathrm{mex}}(0)=1$ and, for $n\\ge1$, $\\overline{σ_o\\mathrm{mex}}(n) \\equiv 0 \\pmod{4}$ if and only if $n$ is a perfect square. We further obtain congruences for the partial sums and self-convolution of $\\overline{σ_o\\mathrm{mex}}(n)$; in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions $d_1(n)$ and $d_3(n)$. We conclude the paper by establishing a Hardy--Ramanujan-type asymptotic formula for $\\overline{σ_o\\mathrm{mex}}(n)$ via the Wright circle method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32207",
  "author_total": 2,
  "authors": [
    {
      "name": "Veena V S"
    },
    {
      "name": "S N Fathima"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b40kkag2fabqk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32207",
  "fulltext_url": null,
  "id": "b40kkag2fabqk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32207",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214735902",
  "pdf_url": "https://arxiv.org/pdf/2609.32207",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b40kkag2fabqk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32207",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32207"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the sum of odd minimal excludants over overpartitions",
  "title_text": "On the sum of odd minimal excludants over overpartitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/b40kkag2fabqk",
  "venue": "arXiv",
  "versions": []
}