Problem 367

Erdős Problem #367: declared status 'open'. Formalized: no. Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\geq 1$,$$\prod_{n\leq m<n+k}B_2(m) \ll n^{2+o(1)}?$$Or perhaps even $\ll_k n^2$? Current best: It would also be interesting to find upper and lower bounds for the analogous product with $B_r$ for $r\geq 3$, where $B_r(n)$ is the $r$-full part of $n$ (that is, the product of prime powers $p^a \mid n$ such that $p^{a+1}\nmid n$ and $a\geq r$). Is it true that, for every fixed $r,k\geq 2$ and $\epsilon>0$,$$\limsup \frac{\prod_{n\leq m<n+k}B_r(m) }{n^{1+\epsilon}}\to\infty?$$van Doorn notes in the comments that for $k\leq 2$ we trivially have$$\prod_{n\leq m<n+k}B_2(m) \ll n^{2},$$but that this fails for all $k\geq 3$, and in fact$$\prod_{n\leq m<n+3}B_2(m) \gg n^{2}\log n$$infinitely often. Prize: no. OEIS: A057521. Tags: number theory.