Let $S_{k}$ be the set of sequences of natural numbers of length $k$.
For a natural number $n$ what is the least $k$ s.t. every element in $S_{k}$ has a subsequence $a_{0},a_{1},...,a_{i}$ for $i \leq k$ for which the sum of terms in the subsequence is divisible by $n$. That is
$\sum_{0\leq j \leq i} a_{j} \equiv 0\pmod n $
A lower bound is easy. $k \geq n$ since $S_{k}$ contains a sequence of $k$ $1$s.
Note that for calculation it suffices to consider the sequences of numbers mod $n$. The calculations I have done suggest $k=n$ which I hope is true since it would give a solution to another problem I'm interested in.