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beautiful functional equation problem

Source: Netherlands TST for BxMO 2017 problem 2

February 1, 2018
functionnumber theoryprime numbers

Problem Statement

Let define a function f:N→Zf: \mathbb{N} \rightarrow \mathbb{Z} such that : i)i)f(p)=1f(p)=1 for all prime numbers pp. ii)ii)f(xy)=xf(y)+yf(x)f(xy)=xf(y)+yf(x) for all positive integers x,yx,y find the smallest n≥2016n \geq 2016 such that f(n)=nf(n)=n
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