AkshajDice

A group of friends named $A_1, A_2, \cdots, A_{2013}$ decide to each play the single-player game AkshajDice. The rules to AkshajDice are as follows

-For $1 \le k \le 2013$ , person $A_k$ has to roll $k$ $2013$ -sided dice numbered $d_1, d_2, \cdots, d_k$

-He wins if for all possible pairs of his dice $d_i$ and $d_j$ , dice $d_j$ rolls a larger number than dice $d_i$ if and only if $j > i$ .

Let the expected value of the number of friends who win be $n$ . If $n$ can be expressed as $\dfrac{a}{b}$ for relatively prime positive integers $a,b$ , find the sum of the greatest prime factors of $a$ and $b$ .