{
  "id": "1201.3168",
  "version": "v1",
  "published": "2012-01-16T07:28:45.000Z",
  "updated": "2012-01-16T07:28:45.000Z",
  "title": "Practical pretenders",
  "authors": [
    "Paul Pollack",
    "Lola Thompson"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Following Srinivasan, an integer n\\geq 1 is called practical if every natural number in [1,n] can be written as a sum of distinct divisors of n. This motivates us to define f(n) as the largest integer with the property that all of 1, 2, 3,..., f(n) can be written as a sum of distinct divisors of n. (Thus, n is practical precisely when f(n)\\geq n.) We think of f(n) as measuring the \"practicality\" of n; large values of f correspond to numbers n which we term practical pretenders. Our first theorem describes the distribution of these impostors: Uniformly for 4 \\leq y \\leq x, #{n\\leq x: f(n)\\geq y} \\asymp \\frac{x}{\\log{y}}. This generalizes Saias's result that the count of practical numbers in [1,x] is \\asymp \\frac{x}{\\log{x}}. Next, we investigate the maximal order of f when restricted to non-practical inputs. Strengthening a theorem of Hausman and Shapiro, we show that every n > 3 for which f(n) \\geq \\sqrt{e^{\\gamma} n\\log\\log{n}} is a practical number. Finally, we study the range of f. Call a number m belonging to the range of f an additive endpoint. We show that for each fixed A >0 and \\epsilon > 0, the number of additive endpoints in [1,x] is eventually smaller than x/(\\log{x})^A but larger than x^{1-\\epsilon}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-16T07:28:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11N37"
    ],
    "keywords": [
      "distinct divisors",
      "generalizes saiass result",
      "practical number",
      "additive endpoint",
      "largest integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.3168P"
    }
  }
}