Numbers in decimal expansions

 Published on Saturday, 25th December 2010, 01:00 pm; Solved by 623;
Difficulty rating: 55%

Problem 316

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
pk, pk+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 $\sum \limits_{n = 2}^{999999} {g \left ( \left \lfloor \dfrac{10^{16}}{n} \right \rfloor \right )}$.

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