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

Jean-Marie De Koninck and Nicolas Doyon

International Journal of Mathematics and Mathematical Sciences, 2003, vol. 2003, 1-8

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 / 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.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:639295

DOI: 10.1155/S016117120320418X

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