Erdős Problem #942

The problem

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 $n$,

$$h(n) >(\log n)^{c-o(1)}?$$

Statement quoted from erdosproblems.com/942, maintained by Thomas Bloom. Original source: [Er76d,p.35]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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