{
  "id": "2105.06190",
  "version": "v2",
  "published": "2021-05-13T11:04:40.000Z",
  "updated": "2021-06-09T16:22:12.000Z",
  "title": "{On terms in a dynamical divisibility sequence having a fixed G.C.D with their index",
  "authors": [
    "Abhishek Jha"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $F(x)$ be an integer coefficient polynomial with degree at least $2.$ Define the sequence $a_n$ by $a_n=F(a_{n-1})$ for all $n\\ge 1$ and $a_0=0.$ Let $\\mathscr{B}_{F,G,k}$ be the set of all positive integers $n$ such that $k\\mid \\gcd(G(n),a_n)$ and if $p\\mid \\gcd(G(n),a_n)$ for some $p$, then $p\\mid k.$ And $\\mathscr{A}_{F,G,k}$ be the subset of $\\mathscr{B}_{F,G,k}$ such that $\\mathscr{A}_{F,G,k}=\\{n>0 : \\gcd(G(n),a_n)=k\\}.$ In this article, we explain the asymptotic density of $\\mathscr{A}_{F,G,k}$ and $\\mathscr{B}_{F,G,k}$ for a class of $(F,G)$ and also compute the explicit density of $\\mathscr{A}_{F,G,k}$ and $\\mathscr{B}_{F,G,k}$ for $G(x)=x.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-06-09T16:22:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C08",
      "11B05",
      "11A05"
    ],
    "keywords": [
      "dynamical divisibility sequence",
      "integer coefficient polynomial",
      "explicit density",
      "asymptotic density",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}