Vol. 72, No. 2, 1977

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ISSN: 0030-8730
On the distribution of some generalized square-full integers

D. Suryanarayana

Vol. 72 (1977), No. 2, 547–555

Let a and b be fixed positive integers. Let n = p1a1p2a2pxax be the canonical representation of n > 1 and let Ra,b denote the set of all n with the property that each exponent ai (1 i r) is either a multiple of a or is contained in the progression at + b, t 0. It is clear that R2,3 = 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 Ra,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 Ra,b(x), the enumerative function of the set Ra,b, in various cases. In this paper, we improve the 0-estimates of the error terms in the asymptotic formulae for Ra,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
Received: 2 February 1977
Revised: 27 March 1977
Published: 1 October 1977
D. Suryanarayana