## GCD of Divisors

### Problem 530

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`(10^{15}).