OPEN
Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that
\[h(n) < (\log n)^{c+o(1)}\]
and, for infinitely many $n$,
\[h(n) >(\log n)^{c-o(1)}?\]
Erdős writes it is not hard to prove that $\limsup h(n)=\infty$, and that the density $\delta_l$ of integers for which $h(n)=l$ exists and $\sum \delta_l=1$.
