Vol. 38, No. 3, 1971

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On the density of (k, r) integers

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

Vol. 38 (1971), No. 3, 613–618

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 = aKb, where a and b are natural numbers and b is r-free. Clearly, a(,r)-integer is a r-free integer. Let Qk,r denote the set of (k,r)-integers and let δ(Qk,r),D(Qk,r) respectively denote the asymptotic and Schnirelmann densities of the set Qk,r. In this paper, we prove that δ(Qk,r) > D(Qk,r) ζ(k)(1 ppr) 1∕k(1 (1∕k))k1, and deduce the known results for r-free integers.

Mathematical Subject Classification
Primary: 10H25
Received: 28 July 1970
Revised: 26 January 1971
Published: 1 September 1971
Y. K. Feng
M. V. Subba Rao