{
  "id": "1810.13063",
  "version": "v1",
  "published": "2018-10-31T01:39:30.000Z",
  "updated": "2018-10-31T01:39:30.000Z",
  "title": "On the number of total prime factors of an odd perfect number",
  "authors": [
    "Joshua Zelinsky"
  ],
  "comment": "54 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N$ be an odd perfect number. Let $\\omega(N)$ be the number of distinct prime factors of $N$ and let $\\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \\frac{302}{113}\\omega - \\frac{286}{113} \\leq \\Omega. $ If $3|N$, then $\\frac{66}{25}\\omega-5\\leq\\Omega.$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $\\omega(N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-31T01:39:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A25"
    ],
    "keywords": [
      "odd perfect number",
      "total prime factors",
      "smallest prime factor",
      "lower bounds",
      "distinct prime factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}