Question 64
Determine which integers $n \gt 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \dots$ of nonzero integers such that the equality
$$
a_k + 2a_{2k} + \dots + n a_{nk} = 0
$$
holds for every positive integer $k$.
$$
a_k + 2a_{2k} + \dots + n a_{nk} = 0
$$
holds for every positive integer $k$.