2012 USAMO 第 3 题

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\dots$ of nonzero integers such that the equality

$$a_k + 2a_{2k} + \dots + na_{nk} = 0$$

holds for every positive integer $k$.

确定哪些整数 $n > 1$ 具有存在非零整数的无限序列 $a_1$、$a_2$、$a_3$、$\dots$ 的属性，使得等式

$$a_k + 2a_{2k} + \dots + na_{nk} = 0$$

对于每个正整数 $k$ 都成立。