On the distribution of some generalized square-full integers

D. Suryanarayana

Vol. 72 (1977), No. 2, 547–555

DOI: 10.2140/pjm.1977.72.547

Abstract

Let a and b be fixed positive integers. Let n = p₁^a₁p₂^a₂⋯p_x^a_x be the canonical representation of n > 1 and let R_a,b denote the set of all n with the property that each exponent a_i (1 ≦ i ≦ r) is either a multiple of a or is contained in the progression at + b, t ≧ 0. It is clear that R_2,3 = L, the set of square-full integers; that is, the set of all n with property that each prime factor of n divides n to at least the second power. Thus the elements of R_a,b may be called generalized square-full integers. This generalization of square-full integers has been given by E. Cohen in 1963, who also established asymptotic formulae for R_a,b(x), the enumerative function of the set R_a,b, in various cases. In this paper, we improve the 0-estimates of the error terms in the asymptotic formulae for R_a,b(x) established by E. Cohen in some cases and further improve them on the assumption of the Riemann hypothesis.

Mathematical Subject Classification

Primary: 10H20, 10H20

Secondary: 10H25

Milestones

Received: 2 February 1977

Revised: 27 March 1977

Published: 1 October 1977