{
  "id": "1906.10001",
  "version": "v1",
  "published": "2019-06-24T14:46:45.000Z",
  "updated": "2019-06-24T14:46:45.000Z",
  "title": "On a problem of de Koninck",
  "authors": [
    "Tomohiro Yamada"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\sigma(n)$ and $\\gamma(n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\\neq 1, 1782$ and $\\sigma(n)=(\\gamma(n))^2$, then there exist odd (not necessarily distinct) primes $p, p^\\prime$ and (not necessarily odd) distinct primes $q_i (i=1, 2, \\ldots, k)$ such that $p, p^\\prime\\mid\\mid n$, $q_i^2\\mid\\mid n (i=1, 2, \\ldots, k)$ and $q_1\\mid \\sigma(p^2), q_{i+1}\\mid\\sigma(q_i^2) (i=1, 2, \\ldots, k-1), p^\\prime \\mid\\sigma(q_k^2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-24T14:46:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "11A05",
      "11A25",
      "11A41"
    ],
    "keywords": [
      "distinct prime divisors",
      "necessarily distinct",
      "necessarily odd"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}