RO-NMO 2014 Problem 102

Let be a function $f:\mathbb{N}\longrightarrow\mathbb{N}$ satisfying $\text{(i)} f(1)=1$ $\text{(ii)} f(p)=1+f(p-1),$ for any prime $p$ $\text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u),$ for any natural number $u$ and any primes $p_1,p_2,\ldots ,p_u.$ Show that $2^{f(n)}\le n^3\le 3^{f(n)},$ for any natural $n\ge 2.$