Practical numbers and the distribution of divisors

Andreas Weingartner

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arXiv · 1405.2585

Practical numbers and the distribution of divisors

Andreas Weingartner

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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 $t\ge 2$.

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Andreas Weingartner. 2014-05-11. Practical numbers and the distribution of divisors. https://arxiv.org/abs/1405.2585

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