I claim $n=2$ is the only possibility for which $(a_1,a_2)=(2,5)$ works. It suffices to rule out $n\ge 3$. Assume the contrary that $n\ge 3$ and $a_1,\dots,a_n\ge 2$ with $a_i\mid a_j^2+1$ for any $i\ne j$. Clearly $a_i$ are pairwise coprime and distinct. Now suppose $a_1=\min\{a_1,\dots,a_n\}$. Using $a_2,a_3\mid a_1^2+1$ and $(a_2,a_3)=1$, we find $a_2a_3\mid a_1^2+1$, while $a_2,a_3\ge a_1+1$, so $(a_1+1)^2 \le a_2a_3\le a_1^2+1$, yielding a contradiction.

grupyorum wrote: I claim $n=2$ is the only possibility for which $(a_1,a_2)=(2,5)$ works. $(2,5,13,17)$ UPD: Sorry, I didn't read the condition properly.