# The constant factor in the asymptotic for practical numbers

## TL;DR

This paper determines the precise constant factor in the asymptotic count of practical numbers, showing it equals approximately 1.33607, refining understanding of their distribution.

## Contribution

The paper calculates the exact constant in the asymptotic formula for practical numbers, improving previous estimates.

## Key findings

- The constant c in the asymptotic is approximately 1.33607.
- Practical numbers are asymptotically distributed as c x / log x.
- The result refines the understanding of practical number distribution.

## 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 $c x/\log x$, where $c$ is a constant. In this note we show that $c=1.33607...$.