{
  "abstract": "Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\\sum_{n\\ge0} a(n)t^n/n! = 1+\\sum_{m\\ge1}\\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $φ(k)$. We prove the stronger statement that the Carmichael function $λ(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\\equiv a(np^{r-1}+i)\\pmod{p^r}$ for every $i\\ge0$, every prime $p$, and all $n,r\\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\\in\\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07238",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahaan Kallat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kerav25mkptd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07238",
  "fulltext_url": null,
  "id": "3kerav25mkptd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07238",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212032386",
  "pdf_url": "https://arxiv.org/pdf/2609.07238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kerav25mkptd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Proof of Bala's Congruence Conjectures for A158690",
  "title_text": "A Proof of Bala's Congruence Conjectures for A158690",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3kerav25mkptd",
  "venue": "arXiv",
  "versions": []
}