BALKAN 2020 Problem 2

Denote $\mathbb{Z}_{>0}=\{1,2,3, \ldots\}$ the set of all positive integers. Determine all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, i) $\sum_{k=1}^{n} f(k)$ is a perfect square, and ii) $f(n)$ divides $n^{3}$.