ÖMO 2015 Problem 1

Let $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ be a function with the following properties:

(i) $f(1) = 0$ (ii) $f(p) = 1$ for all prime numbers $p$ (iii) $f(xy) = y \cdot f(x) + x \cdot f(y)$ for all $x,y$ in $\mathbb{Z}_{>0}$ Determine the smallest integer $n \ge 2015$ that satisfies $f(n) = n$ .

(Gerhard J. Woeginger)