Numbers in decimal expansions

Let p = p₁ p₂ p₃ ... 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.p₁ p₂ p₃ .... 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, ...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 = 31415926535897...., we get k = 9 for p = 355287143650049560000490848764084685354..., we get k = 36 etc and we find that g(535) = 1008.

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

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter