Let $G(a,b)$ be the smallest non-negative integer $n$ for which $\mathbf{gcd}$Greatest common divisor$(n^3+b,(n+a{)}^3+b)$ is maximized.
For example, $G(1,1)=5$ because $gcd(n^3+1,(n+1{)}^3+1)$ reaches its maximum value of $7$ for $n=5$, and is smaller for $0\le n<5$.
Let $H(m,n)=\sum G(a,b)$ for $1\le a\le m$, $1\le b\le n$.
You are given $H(5,5)=128878$ and $H(10,10)=32936544$.

Find $H(18,1900)$.