OPEN

For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that \[Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}.\] Is it true that, for every $\epsilon>0$ and $\ell\geq 1$, if $n$ is sufficiently large then \[Q_2(n(n+1)\cdots(n+\ell))<n^{2+\epsilon}?\] If $\ell\geq 2$ then is \[\limsup_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}\] infinite?

If $\ell\geq 2$ then is \[\lim_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^{\ell+1}}=0?\]

#935: [Er76d]