{
  "id": "1902.09224",
  "version": "v1",
  "published": "2019-02-25T12:22:24.000Z",
  "updated": "2019-02-25T12:22:24.000Z",
  "title": "On the number of distinct exponents in the prime factorization of an integer",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \\#\\{n \\leq x : f(n) = k\\} \\sim A_k x $$ and $$ \\#\\{n \\leq x : f(n) = \\omega(n) - k\\} \\sim \\frac{B x (\\log \\log x)^k}{k! \\log x} , $$ as $x \\to +\\infty$, where $\\omega(n)$ is the number of prime factors of $n$ and $A_k, B > 0$ are some explicit constants. The latter asymptotic extends a result of Akta\\c{s} and Ram Murty about numbers having mutually distinct exponents in their prime factorization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-25T12:22:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11N37",
      "11N64"
    ],
    "keywords": [
      "prime factorization",
      "ram murty",
      "natural number",
      "asymptotic extends",
      "prime factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}