{
  "id": "1309.0906",
  "version": "v8",
  "published": "2013-09-04T03:46:36.000Z",
  "updated": "2015-02-11T22:56:45.000Z",
  "title": "The Abundancy Index of Divisors of Odd Perfect Numbers - Part II",
  "authors": [
    "Keneth Adrian P. Dagal",
    "Jose Arnaldo B. Dris"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "A positive integer $M$ is said to be almost perfect if $\\sigma(M) = 2M - 1$, where $\\sigma(x)$ is the sum of the divisors of $x$. If $N$ is odd and $\\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \\equiv k \\equiv 1 \\pmod 4$ and $\\gcd(q,n) = 1$. In this article, we show that $q^k$, $n$, $qn$ and $n^2$ are not almost perfect. This then implies that the following inequalities hold: $$I(q) \\leq \\frac{2(q - 2)}{q},$$ $$I(n) \\leq \\frac{2n - 2}{n},$$ $$I(qn) \\leq \\frac{2qn - 2}{qn},$$ and $$I(n^2) \\leq \\frac{2n^2 - 253}{n^2},$$ where $I(x) = \\sigma(x)/x$ is the abundancy index of $x$. Lastly, we show that $$I(n^2) < \\frac{2n}{n + 1}$$ if and only if $q < n$.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2013-11-01T04:44:59.000Z",
      "abstract": "A positive integer $M$ is said to be almost perfect if $\\sigma(M) = 2M - 1$, where $\\sigma(x)$ is the sum of the divisors of $x$. If $N$ is odd and $\\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \\equiv k \\equiv 1 \\pmod 4$ and $\\gcd(q,n) = 1$. In this article, we show that $q^k$, $n$, $qn$ and $n^2$ are not almost perfect. This then implies that the following inequalities hold: $$I(q) \\leq \\frac{2(q - 2)}{q},$$ $$I(n) \\leq \\frac{2n - 2}{n},$$ $$I(qn) \\leq \\frac{2qn - 3}{qn},$$ and $$I(n^2) \\leq \\frac{2n^2 - 253}{n^2},$$ where $I(x) = \\sigma(x)/x$ is the abundancy index of $x$.",
      "comment": "18 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v8",
      "updated": "2015-02-11T22:56:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11J25",
      "11J99"
    ],
    "keywords": [
      "odd perfect number",
      "abundancy index",
      "eulerian form",
      "inequalities hold",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.0906D"
    }
  }
}