Consider the numbers from 1 to n, and the random permutation $a_1, a_2, \cdots a_n$. We sum up the permutations in order to get $s_1, s_2, \cdots s_n$ with $s_1=a_1$. The question is, for which $n$ does there exist a permutation such that $n$ is a factor of $s_k$ for all $k$ from 1 to n?
I have tried to find properties of the numbers at each step but nothing has worked. Can someone give a solution?
It only works for $n =1$. The reason being that in order for $s_1$ to be divisible by $n$ you need $a_1 = n$. But then $s_2$ is not divisible by $n$, since $s_2 = a_1 + a_2 = n + a_2$ and $1 \leq a_2 < n$.