Practical pretenders

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}.