Let $p=p_1{p}_2{p}_3\cdots$ be an infinite sequence of random digits, selected from $\{0,1,2,3,4,5,6,7,8,9\}$ with equal probability.
It can be seen that $p$ corresponds to the real number $0.p_1{p}_2{p}_3\cdots$
It can also be seen that choosing a random real number from the interval $[0,1)$ is equivalent to choosing an infinite sequence of random digits selected from $\{0,1,2,3,4,5,6,7,8,9\}$ with equal probability.

For any positive integer $n$ with $d$ decimal digits, let $k$ be the smallest index such that $p_k,p_{k+1},\dots ,p_{k+d-1}$ are the decimal digits of $n$, in the same order.
Also, let $g(n)$ be the expected value of $k$; it can be proven that $g(n)$ is always finite and, interestingly, always an integer number.

For example, if $n=535$, then
for $p=31415926\mathbf{535}897\cdots$, we get $k=9$
for $p=35528714365004956000049084876408468\mathbf{535}4\cdots$, we get $k=36$
etc and we find that $g(535)=1008$.

Given that ${\displaystyle \sum _{n=2}^{999}g\left(\lfloor \frac{{10}^6}{n}\rfloor \right)=27280188}$, find ${\displaystyle \sum _{n=2}^{999999}g\left(\lfloor \frac{{10}^{16}}{n}\rfloor \right)}$.

Note: $\lfloor x\rfloor$ represents the floor function.