Problem 380

Erdős Problem #380: declared status 'proved'. Formalized: no. We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\leq x$ which are contained in at least one bad interval. Is it true that$$B(x)\sim \#\{ n\leq x: P(n)^2\mid n\},$$where $P(n)$ is the largest prime factor of $n$? Current best: 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 $c>0$. Prize: no. OEIS: A070003, A387054, A388654, A389100. Tags: number theory.