JBMO-SL 2021 A3

Let $n$ be a positive integer. A finite set of integers is called $n$ -divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$ -divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$ , or $\{1, 5, 7\}$ and $\{3, 4, 6\}$ . Find all the integers $n > 0$ for which there exists a $n$ -divided set.

Proposed by Martin Rakovsky, France