Question 86
Consider the function $f: \mathbb{N}^* \to \mathbb{N}^*$ which satisfies the properties:
a) $f(1) = 1$,
b) $f(p) = 1 + f(p-1)$ for any prime number $p$,
c) $f(p_1p_2\cdots p_n) = f(p_1) + f(p_2) + \cdots + f(p_n)$ for any prime numbers, not necessary distinct.
Prove that $2^{f(n)} \le n^3 \le 3^{f(n)}$, for any natural number $n$, $n \ge 2$.
a) $f(1) = 1$,
b) $f(p) = 1 + f(p-1)$ for any prime number $p$,
c) $f(p_1p_2\cdots p_n) = f(p_1) + f(p_2) + \cdots + f(p_n)$ for any prime numbers, not necessary distinct.
Prove that $2^{f(n)} \le n^3 \le 3^{f(n)}$, for any natural number $n$, $n \ge 2$.