On the number of prime factors with a given multiplicity over h-free and h-full numbers

Abstract

Let k and n be natural numbers. Let ${\omega }_k\left(n\right)$ denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of ${\omega }_k\left(n\right)$ when restricted to the set of h-free and h-full numbers. We prove that ${\omega }_1\left(n\right)$ has normal order $\log \log n$ over h-free numbers, ${\omega }_h\left(n\right)$ has normal order $\log \log n$ over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ${\omega }_k\left(n\right)$ with $1<k<h$ do not have normal order over h-free numbers and ${\omega }_k\left(n\right)$ with $k>h$ do not have normal order over h-full numbers.