Question 175
We say a positive integer $n$ is "good" if there is a permutation $(a_1, a_2, \dots, a_n)$ of $1, 2, \dots, n$ such that $a_k + k$ is a perfect square for all $1 \le k \le n$. Determine all the good numbers in the set $\{11, 13, 15, 17, 19\}$.