On the density of (k, r) integers

Y. K. Feng and M. V. Subba Rao

Vol. 38 (1971), No. 3, 613–618

DOI: 10.2140/pjm.1971.38.613

Abstract

Let k and r be integers such that 0 < r < k. We call a positive integer n,a(k,r)-integer if it is of the form n = a^Kb, where a and b are natural numbers and b is r-free. Clearly, a(∞,r)-integer is a r-free integer. Let Q_k,r denote the set of (k,r)-integers and let δ(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 δ(Q_k,r) > D(Q_k,r) ≧ ζ(k)(1 −∑ _pp^−r) − 1∕k(1 − (1∕k))^k−1, and deduce the known results for r-free integers.

Mathematical Subject Classification

Primary: 10H25

Milestones

Received: 28 July 1970

Revised: 26 January 1971

Published: 1 September 1971