On the density of (<i>k, r</i>) integers

Abstract

Let ft and r be integers such that 0 < r < ft. We call a positive integer n, a(k f r)-integer if it is of the form n= a b y where a and b are natural numbers and 6 is r-free. Clearly, (oo, r)-integer is a r-free integer. Let Q k , r denote the set of (ft, r)-integers and let (Q k , r ), D(Q k , r ) respectively denote the asymptotic and Schnirelmann densities of the set Q k ,r. In this paper, we prove that (Q k , r ) > D(Q k , r ) (ft)(l - P p-r ) ~ l/ft(l -(I/ft))*" 1 , and deduce the known results for r-free integers.