Is it possible to describe all positive integer sequences ${x_i}$ such that $$ \sum_i x_i=n \quad \hbox{and} \quad \sum_i x_i^k\equiv 0\pmod n \quad \hbox{for all $k$ (and a given $n$)?} $$
Background
We say that two elements of a group belong to the same tribe if their squares are equal. Then
the sum of $k$th powers of tribe sizes is divisible by the order of the group for any positive integer $k$ [http://arxiv.org/abs/1205.2824, Example 1].
This remains true if we replace squares (in the definition of tribes) with cubes or any other powers.
