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

Fix $r\ge2$ 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 $\log(1/D_k)=\frac{r-1}{ζ(r)}k\log k+\frac{r}{ζ(r)}k\log\log k-C_rk+o(k)$, where $C_r=\frac{1}{ζ(r)}((r-1)\logζ(r)+r-1+r\log\fracπ{r\sin(π/r)})$; for $r=2$ this is $C_2=\frac{6}{π