Or as weaker version also works with $M<=N$ (but $M>=\sqrt[3]{N}$, $N$ >1000):
$$\sum_{i=1}^M r_i \equiv 0 \mod N$$
and $$ S = \{s_j = \sum_{i=1}^j r_i \mod N, \forall j<=M\}$$ $$|S|=M$$
The values $R=\{r_i, \forall i\}$ with
$$ r_{i+M}=r_i$$
don't need to pass any tests for randomness. Only
$s_j$ need to be unique values and there is no direct form to compute $r_i$ or $s_j$ with given index or index out of given value (except trivial case $0,1,M$).
Furthermore if a value $s_k$ with unknown $k$ is given there should be no easy to compute function for $s_{k-1}$. That implies same for $r_m$, unknown $m$ and $r_{m-1}$.
Any idea what could work for such and value generator $r$?
(function $r_{n+1}$ independent of $S_n$)