Question 104
Define the function $S: \mathbb{Z}^+ \to \mathbb{Z}^+$ to be the sum of prime divisors (not necessarily distinct) of $n$ (if $v_p(n) = a$ then add $p \cdot a$ to the sum). Prove that
$$
\frac{\tau(n!)}{\tau((n-1)!)} \le \frac{4S(n)}{n}.
$$
$$
\frac{\tau(n!)}{\tau((n-1)!)} \le \frac{4S(n)}{n}.
$$