Problem 890

Erdős Problem #890: declared status 'open'. Formalized: yes. If $\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\geq 1$,$$\liminf_{n\to \infty}\sum_{0\leq i<k}\omega(n+i)\leq k+\pi(k)?$$Is it true that$$\limsup_{n\to \infty}\left(\sum_{0\leq i<k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1?$$ Current best: A question of Erd\H{o}s 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 $k$. Prize: no. Tags: number theory, primes.