Find the smallest $n > 1$ such that the $n$-th prime $p_n \equiv 330 \mod n $.
I was investigating the remainders when the $n$-th prime is divided by $n$. For every positive integer $a < 330$, I have found a prime $p_n$ such that $p_n \equiv a\mod n $. However for $a = 330$, I have not found a solution so far for $n \le 4.5 \times 10^8$. There is no reason to believe why a solution should not exist specifically for 330 so I guess there is a solution which is really large.