Will and Lucas are playing a game. Will claims that he has a polynomial f with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute i for the value of f(i), starting from minute 1. If Will is telling the truth, he will report f(i). Otherwise, he will randomly and uniformly pick a positive integer from the range [1,(i+1)!]. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round 20 is \frac{a}{b}. If the prime factorization of b is p_{1}^{e_{1}} \ldots p_{k}^{e_{k}}, determine the sum \sum_{i=1}^{k} e_{i}.