{
  "id": "0912.2508",
  "version": "v1",
  "published": "2009-12-13T15:10:50.000Z",
  "updated": "2009-12-13T15:10:50.000Z",
  "title": "On the series of the reciprocals lcm's of sequences of positive integers: A curious interpretation",
  "authors": [
    "Bakir Farhi"
  ],
  "comment": "14 pages",
  "journal": "Integers: Electronic Journal of Combinatorial Number Theory, 9 (2009), p. 555-567 (#A42)",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we prove the following result: {quote} Let $\\A$ be an infinite set of positive integers. For all positive integer $n$, let $\\tau_n$ denote the smallest element of $\\A$ which does not divide $n$. Then we have $$\\lim_{N \\to + \\infty} \\frac{1}{N} \\sum_{n = 1}^{N} \\tau_n = \\sum_{n = 0}^{\\infty} \\frac{1}{\\lcm\\{a \\in \\A | a \\leq n\\}} .$${quote} In the two particular cases when $\\A$ is the set of all positive integers and when $\\A$ is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of ${(\\tau_n)}_n$. Furthermore, we discuss the irrationality of the limit of $\\tau_n$ (in the average sense) by applying a result of Erd\\H{o}s.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-13T15:10:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "40A05"
    ],
    "keywords": [
      "positive integer",
      "reciprocals lcms",
      "curious interpretation",
      "average asymptotic behavior",
      "infinite set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2508F"
    }
  }
}