Question 2
Let $n$ be a positive integer. Let $s: \mathbb{N} \to \{1, \dots, n\}$ be a function such that $n$ divides $m - s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \dots$ be a sequence such that $a_0 = 0$ and
$$
a_k = a_{k-1} + s(k) \text{ for all } k \ge 1.
$$
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$.
$$
a_k = a_{k-1} + s(k) \text{ for all } k \ge 1.
$$
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$.