{
  "id": "1405.3587",
  "version": "v3",
  "published": "2014-05-14T17:38:10.000Z",
  "updated": "2015-09-14T17:03:03.000Z",
  "title": "On the number of prime factors of values of the sum-of-proper-divisors function",
  "authors": [
    "Lee Troupe"
  ],
  "comment": "12 pages",
  "journal": "J. Number Theory 150 (2015) pp. 120-135",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\omega(n)$ (resp. $\\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\\omega(n)$ is $\\log\\log n$, and the same is true of $\\Omega(n)$; roughly speaking, a typical natural number $n$ has about $\\log\\log n$ prime factors. We prove a similar result for $\\omega(s(n))$, where $s(n)$ denotes the sum of the proper divisors of $n$: For any $\\epsilon > 0$ and all $n \\leq x$ not belonging to a set of size $o(x)$, \\[ |\\omega(s(n)) - \\log\\log s(n)| < \\epsilon \\log\\log s(n) \\] and the same is true for $\\Omega(s(n))$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-18T13:13:35.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-09-14T17:03:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N56",
      "11N64"
    ],
    "keywords": [
      "prime factors",
      "sum-of-proper-divisors function",
      "proper divisors",
      "normal order",
      "typical natural number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3587T"
    }
  }
}