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 |$c x/\log x$|⁠, 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 |$t\ge 2$|⁠.