{
  "id": "1412.3080",
  "version": "v1",
  "published": "2014-12-09T20:06:18.000Z",
  "updated": "2014-12-09T20:06:18.000Z",
  "title": "On Sparsely Schemmel Totient Numbers",
  "authors": [
    "Colin Defant"
  ],
  "comment": "14 pages, 0 figures, Supported by National Science Foundation grant no. 1262930",
  "categories": [
    "math.NT"
  ],
  "abstract": "For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \\[S_r(p^{\\alpha})=\\begin{cases} 0, & \\mbox{if } p\\leq r; \\\\ p^{\\alpha-1}(p-r), & \\mbox{if } p>r \\end{cases}\\] for all primes $p$ and positive integers $\\alpha$. The function $S_1$ is simply Euler's totient function $\\phi$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $\\phi(n)<\\phi(m)$ for all integers $m>n$. We define a sparsely Schemmel totient number of order $r$ to be a positive integer $n$ such that $S_r(n)>0$ and $S_r(n)<S_r(m)$ for all $m>n$ with $S_r(m)>0$. We then generalize some of the results of Masser and Shiu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-09T20:06:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25"
    ],
    "keywords": [
      "sparsely schemmel totient number",
      "positive integer",
      "eulers totient function",
      "results concerning sparsely totient numbers",
      "schemmel totient function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3080D"
    }
  }
}