{
  "id": "2210.06055",
  "version": "v1",
  "published": "2022-10-12T09:52:25.000Z",
  "updated": "2022-10-12T09:52:25.000Z",
  "title": "On The Tree Structure of Natural Numbers, II",
  "authors": [
    "Vitalii V. Iudelevich"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Each natural number can be associated with some tree graph. Namely, a natural number $n$ can be factorized as $$ n = p_1^{\\alpha_1}\\ldots p_k^{\\alpha_k},$$ where $p_i$ are distinct prime numbers. Since $\\alpha_i$ are naturals, they can be factorized in such a manner as well. This process may be continued, building the \"factorization tree\" until all the top numbers are $1$. Let $H(n)$ be the height of the tree corresponding to the number $n$, and let the symbol $\\uparrow\\uparrow$ denote tetration. In this paper, we derive the asymptotic formulas for the sums $$\\mathcal{M}(x) = \\sum_{p\\leqslant x} H(p-1),\\ \\ \\mathcal{H}(x) = \\sum_{n\\leqslant x}2\\uparrow\\uparrow H(n),$$ and $$\\mathcal{L}(x) = \\sum_{n\\leqslant x}\\dfrac{2\\uparrow\\uparrow H(n)}{2\\uparrow\\uparrow H(n+1)},$$ where the summation in the first sum is taken over primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-12T09:52:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "natural number",
      "tree structure",
      "distinct prime numbers",
      "tree graph",
      "factorization tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}