{
  "id": "2007.03088",
  "version": "v1",
  "published": "2020-07-06T21:55:01.000Z",
  "updated": "2020-07-06T21:55:01.000Z",
  "title": "Arithmetic properties of the sum of divisors",
  "authors": [
    "Tewodros Amdeberhan",
    "Victor H. Moll",
    "Vaishavi Sharma",
    "Diego Villamizar"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The divisor function $\\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \\in \\mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for $\\nu_{p}(\\sigma(n))$ are established. For $p=2$, these involve only the odd primes dividing $n$. These expressions are used to establish the bound $\\nu_{2}(\\sigma(n)) \\leq \\lceil\\log_{2}(n) \\rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is $\\nu_{p}(\\sigma(n)) \\leq \\lceil \\log_{p}(n) \\rceil$, with equality related to solutions of the Ljunggren-Nagell diophantine equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-06T21:55:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11D61",
      "11A41"
    ],
    "keywords": [
      "arithmetic properties",
      "odd prime",
      "ljunggren-nagell diophantine equation",
      "distinct mersenne primes",
      "divisor function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}