The density of long runs of non-$r$-free integers

Xinyang Jiang

Fix $r\ge 2$ and let $D_k$ be the natural density of the $n$ such that none of $n+1,\dots ,n+k$ is $r$-free. We prove that $\log(1/D_k)=\frac{r-1}{ζ(r)}k\log k+\frac{r}{ζ(r)}k\log\lo…