The number $209$ can be expressed as $a^2+3ab+b^2$ in two distinct ways:

$209=8^2+3\cdot 8\cdot 5+5^2$
$209={13}^2+3\cdot 13\cdot 1+1^2$

Let $f(n,r)$ be the number of integers $k$ not exceeding $n$ that can be expressed as $k=a^2+3ab+b^2$, with $a>b>0$ integers, in exactly $r$ different ways.

You are given that $f({10}^5,4)=237$ and $f({10}^8,6)=59517$.

Find $f({10}^{15},40)$.