Problem 343

Determine the smallest integer $n \geqslant 2$ such that there exist strictly positive integers $\left(a_{i}\right)_{1 \leqslant i \leqslant n}$ satisfying:

$$\sum_{i=1}^{n} a_{i}^{2} \mid\left(\sum_{i=1}^{n} a_{i}\right)^{2}-1$$