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Problem #397

Factorial divisibility
 Public ★(x10) 05/27/17 by Philippe_57721 8xp Programming 83.3%

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.

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