{
  "id": "1605.07424",
  "version": "v1",
  "published": "2016-05-24T12:43:41.000Z",
  "updated": "2016-05-24T12:43:41.000Z",
  "title": "On the p-adic valuation of Stirling numbers of the first kind",
  "authors": [
    "Paolo Leonetti",
    "Carlo Sanna"
  ],
  "comment": "12 pages, 3 figures",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For all integers $n \\geq k \\geq 1$, define $H(n,k) := \\sum 1 / (i_1 \\cdots i_k)$, where the sum is extended over all positive integers $i_1 < \\cdots < i_k \\leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erd\\H{o}s-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \\geq 2$, and $x \\geq (k-1)p$, we prove that $\\nu_p(H(n,k)) < -(k - 1)(\\log_p(n/(k - 1)) - 1)$ for all positive integers $n \\in [(k-1)p, x]$ whose base $p$ representations start with the base $p$ representation of $k - 1$, but at most $3x^{0.835}$ exceptions. We also generalize a result of Lengyel by giving a description of $\\nu_2(H(n,2))$ in terms of an infinite binary sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-24T12:43:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B73",
      "11B50",
      "11A51"
    ],
    "keywords": [
      "first kind",
      "stirling numbers",
      "p-adic valuation",
      "positive integers",
      "infinite binary sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}