Square-full numbers multiple of a certain set of primes and hybrid numbers

Abstract

Let h ≥ 1 be an arbitrary but fixed positive integer. Let us consider the s distinct primes q₁, . . . , q_s. Let B_q1···qs(x) be the number of h-full numbers not exceeding x multiple of q₁ · · · q_s and let A_q1···qs (x) be the number of h-full numbers not exceeding x relatively prime to q₁ · · · q_s. In this note we obtain asymptotic formulas for A_q1···qs (x) and B_q1···qs (x). Then we apply the results obtained in the study of hybrid h-full numbers. That is, the h-full numbers of the form q^hQ, where q > 1 is square-free, Q > 1 is a (h + 1)-full number, and q, Q are relatively prime.