Erdős Problem #367

The problem

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 $\ll_k n^2$?

Statement quoted from erdosproblems.com/367, maintained by Thomas Bloom. Original source: [ErGr80,p.68]. 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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