Problem 935

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?

Status

Open

No claim settles this problem.