Dasith the number wizard can manipulate numbers very quickly. For his greatest trick, he asks his audience for any positive integer $p$. He then pulls out of his hat 3 integers $x$, $y$, and $z$, with $0 < x^2 + y^2 + z^2 < p^2$, such that $p | (x^2 + y^2 + z^2)$.
This trick is possible for some choices of $p$, and impossible for others. Find any infinite subset of the positive integers such that this trick is possible for any choice of $p$ in that set.