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International Journal of Mathematics and Mathematical Sciences
Volume 22, Issue 3, Pages 655-658
http://dx.doi.org/10.1155/S0161171299226555

On a density problem of Erdös

Department of Mathematics, Hofstra University, Hempstead 11550, NY, USA

Received 13 April 1998; Revised 10 June 1998

Copyright © 1999 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.

Abstract

For a positive integer n, let P(n) denotes the largest prime divisor of n and define the set: 𝒮(x)=𝒮={nx:ndoes not divideP(n)!}. Paul Erdös has proposed that |S|=o(x) as x, where |S| is the number of nS. This was proved by Ilias Kastanas. In this paper we will show the stronger result that |S|=O(xe1/4logx).