{
  "id": "2208.02242",
  "version": "v1",
  "published": "2022-08-03T17:49:03.000Z",
  "updated": "2022-08-03T17:49:03.000Z",
  "title": "Patterns in the iteration of an arithmetic function",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $S$ be an arithmetic function and let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m$ has increasing-decreasing pattern $V$ with respect to $S$ if, for odd $i$, \\[ S^{v_1+ \\cdots + v_{i-1}}(m) < S^{v_1+ \\cdots + v_{i-1}+1}(m) < \\cdots < S^{v_1+ \\cdots + v_{i-1}+v_{i}}(m) \\] and, for even $i$, \\[ S^{v_1+ \\cdots + v_{i-1}}(m) > S^{v_1+ \\cdots +v_{i-1}+1}(m) > \\cdots > S^{v_1+ \\cdots +v_{i-1}+v_i}(m). \\] The arithmetic function $S$ is wildly increasing-decreasing if, for every finite sequence $V$ of positive integers, there exists an integer $m$ such that $m$ has increasing-decreasing pattern $V$ with respect to $S$. This paper gives a new proof that the Collatz function is wildly increasing-decreasing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-03T17:49:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11B83",
      "11D04"
    ],
    "keywords": [
      "arithmetic function",
      "finite sequence",
      "positive integers",
      "collatz function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}