China Girls' Mathematical Olympiad

The good numbers are $13$, $15$, $17$ and $19$. However $11$ is not.

Note that for $1\le k\le 11$, $4+k$ is a perfect square if and only if $k=5$. Likewise, $11+k$ is a perfect square if and only if $k=5$. Hence $11$ is not good.

Note that $13$ is good because

$k$: 1 2 3 4 5 6 7 8 9 10 11 12 13, $a_k$: 8 2 13 12 11 10 9 1 7 6 5 4 3.

Similarly, $15$, $17$ and $19$ are good because

$k$: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15, $a_k$: 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1;

$k$: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17, $a_k$: 3 7 6 5 4 10 2 17 16 15 14 13 12 11 1 9 8;

$k$: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19, $a_k$: 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9 19 18 17.