International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2003 / Article

Open Access

Volume 2003 |Article ID 639295 | https://doi.org/10.1155/S016117120320418X

Jean-Marie De Koninck, Nicolas Doyon, "On a thin set of integers involving the largest prime factor function", International Journal of Mathematics and Mathematical Sciences, vol. 2003, Article ID 639295, 8 pages, 2003. https://doi.org/10.1155/S016117120320418X

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

Received22 Apr 2002

Abstract

For each integer n2, let P(n) denote its largest prime factor. Let S:={n2:n does not divide P(n)!} and S(x):=#{nx:nS}. 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(xexp{(1/4)logx}). In this paper, we show that S(x)=xexp{(2+o(1))×logxloglogx}. We also investigate small and large gaps among the elements of S and state some conjectures.

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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