On the number of $k$-full integers between three successive $k$-th powers
Shusei Narumi, Yohei Tachiya
TL;DR
The paper addresses the distribution of $k$-full integers between three successive $k$-th powers by introducing a framework based on the canonical representation $n=a^k\lambda^k$ with $\lambda$ drawn from the set $\Lambda_k$. It defines counting sets $\mathcal{A}_{\ell,m}^{(k)}$ tracking the numbers of $k$-full integers in the two adjacent $k$-power gaps and proves explicit positive asymptotic densities for these sets via a decomposition into $\mathcal{B}_{\mathcal{I,J}}^{(k)}$, whose densities factor as $d_{\mathcal{I,J}}^{(k)}=\prod_{\lambda\in\mathcal{I}\cup\mathcal{J}}(1/\lambda)\prod_{\lambda\in\Lambda_k\setminus(\mathcal{I}\cup\mathcal{J})}(1-2/\lambda)$. A generating function $\sum_{\ell,m\ge0} d(\mathcal{A}_{\ell,m}^{(k)}) z^\ell w^m=\prod_{\lambda\in\Lambda_k}(1+(z+w-2)/\lambda)$ then follows, enabling explicit formulas and numerical densities for small $k$ (notably $k=2,3$). The results yield that $\mathcal{A}_{\ell,m}^{(k)}$ have positive densities and imply infinitely many triples of consecutive $k$-th powers within the $k$-full sequence, thereby generalizing Shiu’s square case. The paper also provides exact expressions and computations for the densities, including maxima for small $k$ and discussion of asymptotic additivity across disjoint unions.
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.