SOLVED

Let $t_k(n)$ denote the least $m$ such that
\[n\mid (m+1)(m+2)\cdots (m+k).\]
Is it true that
\[\sum_{1\leq n\leq N}t_2(n)=o(N)?\]

The answer is yes, proved by Hall. It is probably true that the sum is $o(N/(\log N)^c)$ for some constant $c>0$. Similar questions can be asked for other $k\geq 3$.