Factorial divisibility

Let $S(n)$ be the smallest integer such as $n$ divides $S(n)!$.

For instance $S(9)=6$ for $6!\equiv 0\text{ mod }9$

Consider the set $A$ of numbers of form $2^{e_1}\times 3^{e_2}\times 5^{e_3}\times 7^{e_4}\times {11}^{e_5}\times {13}^{e_6}\times {17}^{e_7}\times {19}^{e_8}$

What is $\sum _{n\in A^{\prime }}S(n)$ where $A^{\prime }$ contains the first ${10}^8$ elements of A

You are given : 45499522 when $A^{\prime }$ contains the first ${10}^6$ elements of A

[My timing : 120 sec]

P.S1: We consider S(1) = 0
P.S2: There is a solution in less than 15 sec.