On the number of $k$-full integers between three successive $k$-th powers

Abstract

Let $k\geq2$ be an integer. The aim of this paper is to investigate the distribution of $k$-full integers between three successive $k$-th powers. More precisely, for any integers $\ell,m\ge0$, we establish the explicit asymptotic density for the set of integers $n$ such that the intervals $(n^k, (n+1)^k)$ and $((n+1)^k, (n+2)^k)$ contain exactly $\ell$ and $m$ $k$-full integers, respectively. As an application, we prove that there are infinitely many triples of successive $k$-th powers in the sequence of $k$-full integers, thereby providing a more general answer to Shiu's question.