On a thin set of integers involving the largest prime factor function. (1st April 2003) - De Koninck, Jean-Marie; Doyon, Nicolas
Abstract : For each integern ≥ 2, letP ( n ) denote its largest prime factor. LetS : = { n ≥ 2 : n does not divideP ( n ) ! } andS ( x ) : = # { n ≤ x : n ∈ S } . Erdős (1991) conjectured thatS is a set of zero density. This was proved by Kastanas (1994) who established thatS ( x ) = O ( x / log x ) . Recently, Akbik (1999) proved thatS ( x ) = O ( x   exp { − ( 1 / 4 ) log x } ) . In this paper, we show thatS ( x ) = x   exp { − ( 2 + o ( 1 ) ) × log   x   log   log   x } . We also investigate small and large gaps among the elements ofS and state some conjectures.