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Problem 1 -- Punctual Functional

Source: 46th Austrian Mathematical Olympiad National Competition Part 2 Problem 1

July 14, 2018
Austriaalgebrafunctional equation

Problem Statement

Let f:Z>0→Zf: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} be a function with the following properties:
(i) f(1)=0f(1) = 0 (ii) f(p)=1f(p) = 1 for all prime numbers pp (iii) f(xy)=y⋅f(x)+x⋅f(y)f(xy) = y \cdot f(x) + x \cdot f(y) for all x,yx,y in Z>0\mathbb{Z}_{>0}
Determine the smallest integer n≥2015n \ge 2015 that satisfies f(n)=nf(n) = n.
(Gerhard J. Woeginger)
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