2008 IMO Shortlist C2

For every positive integer $n$ determine the number of permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of the set $\{1,2, \ldots, n\}$ with the following property: $$2\left(a_{1}+a_{2}+\cdots+a_{k}\right) \quad \text { is divisible by } k \text { for } k=1,2, \ldots, n \text {. }$$

对于每个正整数 $n$，确定集合 $\{1,2, \ldots, n\}$ 的排列数 $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$，其属性如下： $$2\left(a_{1}+a_{2}+\cdots+a_{k}\right) \quad \text { 可被 } k \text { 整除对于 } k=1,2, \ldots, n \text {. }$$