International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 19, Pages 1185-1192
doi:10.1155/S016117120320418X
On a thin set of integers involving the largest prime factor function
Jean-Marie De Koninck1
and Nicolas Doyon2
1Département de Mathématiques et de Statistique, Université Laval, Québec, Québec G1K 7P4, Canada
2Département de Mathématiques et de Statistique, Université de Montréal, Québec, Montréal H3C 3J7, Canada
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.