Factorial divisibility
Public  05/27/17  8xp  Programming  84.6% 
Let $S(n)$ be the smallest integer such as $ n $ divides $S(n)!$.
For instance $S(9) = 6$ for $ 6! \equiv 0 \textrm{ 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 $\sum\limits_{n \in A^\prime} S(n)$ where $ A' $ contains the first $10^8 $ elements of A
You are given : 45499522 when $ A' $ 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.
For instance $S(9) = 6$ for $ 6! \equiv 0 \textrm{ 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 $\sum\limits_{n \in A^\prime} S(n)$ where $ A' $ contains the first $10^8 $ elements of A
You are given : 45499522 when $ A' $ 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.
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