For which positive integers does the following hold: If \(a_1, a_2, \ldots, a_n\) are positive integers, \(a_k \leq n\) for each \(k\) and \(\sum_{k=1}^n=2 n\), then it is always possible to choose \(a_{i_1}, a_{i_2}, \ldots, a_{i_j}\) so that indexes \(i_1, i_2, \ldots, i_j\) are different numbers and \(\sum_{k=1}^j a_{i_k}=n\) ?