Integers with large practical component

TL;DR

This paper investigates the distribution of integers with large practical divisors, providing an asymptotic estimate for their count up to a given limit, which advances understanding of practical numbers in number theory.

Contribution

It introduces an asymptotic estimate for the number of integers up to x that possess a practical divisor at least y, expanding knowledge on practical numbers and their distribution.

Findings

Derived an asymptotic formula for integers with large practical divisors

Quantified the density of practical numbers within a range

Enhanced understanding of the structure of practical numbers

Abstract

A positive integer $n$ is called practical if all integers between $1$ and $n$ can be written as a sum of distinct divisors of $n$. We give an asymptotic estimate for the number of integers $\le x$ which have a practical divisor $\ge y$.