Solution:
The statement is
$$\frac{n(n+1)}{2} \mid \frac{n^{2}(n^{2}+1)}{2} \Leftrightarrow n+1 \mid n(n^{2}+1) = n^{3}+n.$$
But $n+1$ also divides $(n+1)(n^{2}-n+2) = n^{3}+n+2$, so $n+1$ must divide $2$. Hence, $n$ cannot be greater than $1$. And $n=1$ clearly works, so that is the answer.