Problem Statement
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 $n$,\[h(n) >(\log n)^{c-o(1)}?\]
Categories:
Number Theory Powerful
Progress
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 $\sum \delta_l=1$.A proof that $h(n)$ is unbounded is provided by van Doorn in the comments.
De Koninck and Luca [DeLu04] have proved, for infinitely many $n$,\[h(n) \gg \left(\frac{\log n}{\log\log n}\right)^{1/3}.\]They also give the density ($\approx 0.275$) of those $n$ such that $h(n)=1$.
Source: erdosproblems.com/942 | Last verified: January 19, 2026