$$A=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{3^i{10}^{3^i}}$$

Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward. For example, $A(100)=4938271604$ and $A({10}^8)=2584642393$.

Find $A({10}^{16})$.