On a thin set of integers involving the largest prime factor function

Abstract

For each integer n≥2, let P(n) denote its largest prime factor. Let S:={n≥2:n does not divide P(n)!} and S(x):=#{n≤x:n∈S}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x)=O(x/logx). Recently, Akbik (1999) proved that S(x)=O(x exp{−(1/4)logx}). In this paper, we show that S(x)=x exp{−(2+o(1))×log x log log x}. We also investigate small and large gaps among the elements of S and state some conjectures.