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 / log x ) . Recently, Akbik (1999) proved that S ( x ) = O ( x   exp { − ( 1 / 4 ) log x } ) . 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.