Four. (50 points) Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1599$, when $k^{2}<n \leqslant(k+1)^{2}$, $a_{n}=\left[\frac{k+1}{2}\right]+\left[\frac{k}{2}\right]+1\left(k \in \mathbf{N}^{*}\right)$, does there exist a positive integer $n$ such that the equation $\sum_{i=1}^{n} a_{i}=2006$ holds? If it exists, find all such numbers $n$; if not, explain the reason.