Practical pretenders

TL;DR

This paper introduces a new function f(n) to measure the 'practicality' of integers based on their divisors, analyzes its distribution, maximal order, and the range of values it can take, extending previous results on practical numbers.

Contribution

It defines and studies the function f(n), providing distribution results, bounds on its maximal order for non-practical numbers, and characterizing the range of f(n).

Findings

Distribution of n with f(n) ≥ y is proportional to x / log y.

If f(n) exceeds a certain bound, n must be practical.

Number of additive endpoints grows slower than x / (log x)^A but faster than x^{1-ε}.

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}.