Problem #367: Let $B_2(n)$ be the 2-full part of $n$ (that is,...

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Progress

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.