{
  "abstract": "Let $cϕ_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\\geq2$ and every $k\\equiv2\\pmod{2^m}$, \\[ \\sum_{n\\geq0}cϕ_k(n)q^n\\equiv\\frac{φ(q)\\,(q^2;q^2)_\\infty}{(q;q)_\\infty^2}\\sum_{n\\geq0}cϕ_{k/2}(n)q^{2n}\\pmod{2^m}, \\] where $φ(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $cϕ_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \\[ \\sum_{n\\geq0}cϕ_{18}(6n+1)q^n \\equiv4\\sum_{r\\in\\mathbb{Z}}q^{r(3r-1)/2}\\pmod{16}, \\] which proves the congruences $cϕ_{18}(30n+19)\\equiv cϕ_{18}(30n+25)\\equiv0\\pmod{16}$ recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set $k=10$ and determine $cϕ_{10}(2n+1)$ modulo $8$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11813",
  "author_total": 1,
  "authors": [
    {
      "name": "Manjil P. Saikia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jv82m04npej3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11813",
  "fulltext_url": null,
  "id": "3jv82m04npej3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11813",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212237758",
  "pdf_url": "https://arxiv.org/pdf/2609.11813",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jv82m04npej3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11813",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11813"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generalized Frobenius Partitions Modulo Powers of $2$",
  "title_text": "Generalized Frobenius Partitions Modulo Powers of $2$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/3jv82m04npej3",
  "venue": "arXiv",
  "versions": []
}