Practical numbers and the distribution of divisors

Abstract

An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $cx/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.

Minor improvement of Theorems 2 and 4. To appear in Q. J. Math