ELMO SL 2014 N8

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$ . (ii) For all sufficiently large integers $n$ , we have $f(n) \neq 1$ and $$f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}}$$ for all positive integers $a$ and $b$ .

Proposed by Yang Liu