Erdős and Graham only knew that $B(x) > x^{1-o(1)}$. Similarly, we call an interval $[u,v]$ 'very bad' if $\prod_{u\leq m\leq v}m$ is powerful. The number of integers $n\leq x$ contained in at least one very bad interval should be $\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers $\leq x$.

We have\[\#\{ n\leq x: P(n)^2\mid n\}=\frac{x}{\exp((c+o(1))\sqrt{\log x\log\log x})}\]for some constant $c>0$.

Tao notes in the comments that if $[u,v]$ is bad then it cannot contain any primes, and hence certainly $v<2u$, and in general $v-u$ must be small (for example, assuming Cramer's conjecture, $v-u\ll (\log u)^2$).

See also [382].

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