© Hindawi Publishing Corp. ON A THIN SET OF INTEGERS INVOLVING THE LARGEST PRIME FACTOR FUNCTION

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) con-jectured 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(xexp{−(1/4)√logx}). In this paper, we show that S(x) = xexp{−(2+o(1))×√ logx log logx}. We also investigate small and large gaps among the elements of S and state some conjectures. 2000 Mathematics Subject Classification: 11B05, 11N25