MathNet 2013 Question 5

Find all increasing functions $f: \mathbb{N}^* \to \mathbb{N}^*$ with the property: for every $n \in \mathbb{N}^*$, the number $\frac{1}{n} \sum_{k=1}^{n} k f(k)$ is a perfect square not larger than $(n+1)^2$.