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Problem 484

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Arithmetic Derivative

The arithmetic derivative is defined by

• p’ = 1 for any prime p
• (ab)’ = a’b + ab’ for all integers a, b (Leibniz rule)

For example, 20’ = 24.

Find ∑ gcd(k,k’) for 1 < k ≤ 5·10¹⁵.

Note: gcd(x,y) denotes the greatest common divisor of x and y.

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算术导数

算术导数是按如下方式定义的：

• 对任意素数p，p’ = 1
• 对任意整数a和b，(ab)’ = a’b + ab’（莱布尼兹法则）

例如，20’ = 24。

对于1 < k ≤ 5·10¹⁵，求∑ gcd(k,k’)。

注，gcd(x,y)指x和y的最大公约数.

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