OSN 2015 Problem 2

For every natural number $a$ and $b$ , define the notation $[a,b]$ as the least common multiple of $a$ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$ . Find all $n \in \mathbb{N}$ that satisfies $$4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}$$