OBM 2012 Problem 4

There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that

$$n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}}$$

where $p_i$ is the i-th prime ( $p_1=2,p_2=3,p_3=5,p_4=7,\ldots$ ) and $a_i>1$ for all $i$ ?