Practical numbers and the distribution of divisors

TL;DR

This paper investigates practical numbers, proving their count below x is asymptotic to c x / log x, and estimates the distribution of integers with bounded divisor ratios, confirming conjectures and extending understanding of divisor structures.

Contribution

It proves the asymptotic density of practical numbers and provides a uniform estimate for integers with limited ratios of consecutive divisors, advancing divisor theory.

Findings

Practical numbers below x are asymptotically c x / log x.

Established a uniform estimate for integers with maximum divisor ratio at most t.

Confirmed Margenstern's conjecture on practical number distribution.

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 $t\ge 2$.