KöMaL 2013 A587

Find all positive integers $n$ for which there exists a sequence $a_1$<$a_2$<...<$a_n$ of positive integers with the property that the sums $a_i$+$a_j$ with 1$\le i$<$j \le n$ are distinct and among them each modulo 4 residue occurs the same number of times.