Question 43
Denote $\mathbb{Z}_{\gt 0} = \{1, 2, 3, \dots\}$ the set of all positive integers. Determine all functions $f : \mathbb{Z}_{\gt 0} \to \mathbb{Z}_{\gt 0}$ such that, for each positive integer $n$:
i) $\sum_{k=1}^{n} f(k)$ is a perfect square;
ii) $f(n)$ divides $n^3$.
i) $\sum_{k=1}^{n} f(k)$ is a perfect square;
ii) $f(n)$ divides $n^3$.