To be frank with you, this problem is a classic problem of tremendous magnitude which may increase the difficulty of this problem.
We define a type of operation concerning a positive integer $n$ $(n > 1)$ as to replace it with an integer $d$, one of factors of $n$ $(1 \leq d \leq n)$.
You are given a positive integer $n$ and then we will ask you to determine the expectation number of times to utilize this type of operation if we want to change $n$ into $1$ by operating again and again, assuming each possible $d$ in each operation has equal possibility to select.
For the sake of calculation, $n$ and all its distinct prime factors $p_1, p_2, \cdots, p_m$ will be given, satisfying $n$ has $m$ distinct prime factors exactly.