For integers $b > a > 1$ define $F(a,b)$ to be the largest integer $n$ such that when $n$ is expressed in base $a$ the result is identical to the result when $2n$ is expressed in base $b$. If there is no such integer then $F(a,b) = 0$.

For example, $53 = 1222_3$ and $2\times 53 = 106 = 1222_4$. In addition there is no other integer $n$ that has this property. Hence $F(3,4) = 53$. Also you are given $F(9,10) = 8152650$.

Define $\displaystyle G(a) = \sum_{b=a+1}^{\infty} F(a,b)$. It can be shown that $G(a)$ is always finite. For example $G(3) = 72$ because the only non zero values are $F(3,4) = 53, F(3,6) = 6, F(3,7) = 8, F(3,8) = 5$.

Find $\displaystyle \sum_{a=2}^{20} G(a)$.