The constant factor in the asymptotic for practical numbers

Abstract

An integer $n\ge 1$ is said to be practical if every natural number $m\le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $cx/\log x$, where $c$ is a constant. In this note we show that $c=\mathrm{1.33607...}$.

10 pages