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Problem 530

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GCD of Divisors

Every divisor d of a number n has a complementary divisor n/d.

Let f(n) be the sum of the greatest common divisor of d and n/d over all positive divisors d of n, that is $f(n)={\displaystyle \sum _{d|n}\text{gcd}(d,\frac{n}{d})}$.

Let F be the summatory function of f, that is $F(k)={\displaystyle \sum _{n=1}^{k}f(n)}$.

You are given that F(10)=32 and F(1000)=12776.

Find F(10¹⁵).

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约数的最大公约数

数n的每个约数d都有一个互补约数n/d。

对于数n的所有正约数d，记f(n)是d与n/d的最大公约数之和，即$f(n)={\displaystyle \sum _{d|n}\text{gcd}(d,\frac{n}{d})}$。

记F是f的和函数，即$F(k)={\displaystyle \sum _{n=1}^{k}f(n)}$。

已知F(10)=32以及F(1000)=12776。

求F(10¹⁵)。

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