CGMO 2004 Problem 1

We say a positive integer $n$ is good if there exists a permutation $a_1, a_2, \ldots, a_n$ of $1, 2, \ldots, n$ such that $k + a_k$ is perfect square for all $1\le k\le n$ . Determine all the good numbers in the set $\{11, 13, 15, 17, 19\}$ .