Problem 530
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 $f(n)=\displaystyle\sum\limits_{d|n} \text{gcd}(d,\frac n d)$.
Let F be the summatory function of f, that is $F(k)=\displaystyle\sum\limits_{n=1}^k f(n)$.
You are given that F(10)=32 and F(1000)=12776.
Find F(1015).
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$f(n)=\displaystyle\sum\limits_{d|n} \text{gcd}(d,\frac n d)$。
记F是f的和函数，即$F(k)=\displaystyle\sum\limits_{n=1}^k f(n)$。
已知F(10)=32以及F(1000)=12776。
求F(1015)。