Eccoci.

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