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.