Erdős Problem #935

declared status 'open'. Formalized: no. 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$$\lim_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^{\ell+1}}=0?$$ Current best: A result of Mahler implies, for every $\ell\geq 1$,$$\limsup_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}\geq 1.$$All these questions can be asked replacing $Q_2$ by $Q_r$ for $r>2$, only keeping those prime powers with exponent $\geq r$. Prize: no. OEIS: A057521, A389244. Tags: number theory.

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