Bounds for the counting function of the Jordan-P\'olya numbers

TL;DR

This paper establishes non-trivial bounds on the count of Jordan-Pólya numbers, which are integers expressible as products of factorials, up to a given limit, enhancing understanding of their distribution.

Contribution

It provides the first non-trivial lower and upper bounds for the counting function of Jordan-Pólya numbers, advancing the theoretical understanding of their distribution.

Findings

Derived explicit bounds for the counting function

Improved previous estimates on Jordan-Pólya numbers

Enhanced understanding of factorial product integers

Abstract

A positive integer $n$ is said to be a Jordan-P\'olya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-P\'olya numbers not exceeding a given number $x$.