{
  "id": "2201.09287",
  "version": "v1",
  "published": "2022-01-23T15:07:35.000Z",
  "updated": "2022-01-23T15:07:35.000Z",
  "title": "Numbers of the form $kf(k)$",
  "authors": [
    "Mikhail R. Gabdullin",
    "Vitalii V. Iudelevich"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a function $f\\colon \\mathbb{N}\\to\\mathbb{N}$, define $N^{\\times}_{f}(x)=\\#\\{n\\leq x: n=kf(k) \\mbox{ for some $k$} \\}$. Let $\\tau(n)=\\sum_{d|n}1$ be the divisor function, $\\omega(n)=\\sum_{p|n}1$ be the prime divisor function, and $\\varphi(n)=\\#\\{1\\leq k\\leq n: (k,n)=1 \\}$ be Euler's totient function. We prove that \\begin{gather*} \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! 1) \\quad N^{\\times}_{\\tau}(x) \\asymp \\frac{x}{(\\log x)^{1/2}}; \\\\ 2) \\quad N^{\\times}_{\\omega}(x) = (1+o(1))\\frac{x}{\\log\\log x}; \\\\ \\!\\!\\!\\!\\!\\!\\!\\!\\! 3) \\quad N^{\\times}_{\\varphi}(x) = (c_0+o(1))x^{1/2}, \\end{gather*} where $c_0=1.365...$\\,.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-23T15:07:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eulers totient function",
      "prime divisor function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}