Erdős problem 890

A question of Erdős and Selfridge [ErSe67], who observe that $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\geq k+\pi(k)-1$ for every $k$. This follows from Pólya's theorem that the set of $k$-smooth integers has unbounded gaps - indeed, $n(n+1)\cdots (n+k-1)$ is divisible by all primes $\leq k$ and, provided $n$ is large, all but at most one of $n,n+1,\ldots,n+k-1$ has a prime factor $>k$ by Pólya's theorem.