Determine all integers \(n\geq 2\) having the following property: for any integers \(a_1,a_2,\ldots, a_n\) whose sum is not divisible by \(n\), there exists an index \(1 \leq i \leq n\) such that none of the numbers \[a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}\]is divisible by \(n\). Here, we let \(a_i=a_{i-n}\) when \(i >n\).
Here's something that fails: \(n = 6\) and the sequences \(121212\), \(030303\). In general \(pq\) fails for \(p \neq q\), because you can have something that sums to