{
  "id": "2303.11974",
  "version": "v1",
  "published": "2023-03-21T16:07:38.000Z",
  "updated": "2023-03-21T16:07:38.000Z",
  "title": "On inequalities involving counts of the prime factors of an odd perfect number",
  "authors": [
    "Graeme Clayton",
    "Cody S. Hansen"
  ],
  "comment": "17 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 (counting multiplicity) of prime factors of $N$. We prove that $\\frac{99}{37}\\omega(N) - \\frac{187}{37} \\leq \\Omega(N)$ and that if $3\\nmid N$, then $\\frac{51}{19}\\omega(N)-\\frac{46}{19} \\leq \\Omega(N)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-21T16:07:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11A51",
      "11N32"
    ],
    "keywords": [
      "odd perfect number",
      "inequalities",
      "distinct prime factors",
      "total number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}