On a density problem of Erdös

Abstract

For a positive integer n, let P(n) denotes the largest prime divisor of n and define the set: 𝒮(x)=𝒮={n≤x:n   does not divide   P(n)!}. Paul Erdös has proposed that |S|=o(x) as x→∞, where |S| is the number of n∈S. This was proved by Ilias Kastanas. In this paper we will show the stronger result that |S|=O(xe−1/4logx).