{
  "id": "2101.11585",
  "version": "v1",
  "published": "2021-01-27T18:22:48.000Z",
  "updated": "2021-01-27T18:22:48.000Z",
  "title": "The number of prime factors of integers with dense divisors",
  "authors": [
    "Andreas Weingartner"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \\log \\log n$, where $C=(1-e^{-\\gamma})^{-1} = 2.280...$ and $\\gamma$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-27T18:22:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11N37"
    ],
    "keywords": [
      "prime factors",
      "dense divisors",
      "eulers constant",
      "normal order",
      "arbitrary constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}