Problem

Source: Romania National Olympiad 2014, Grade X, Problem 2

Tags: function



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. $