Let $E(n)$ be the expected value of the index of finding an n-digit number in the digits of pi
e.g.
- Number 0 is found at index 32
- Number 1 is found at index 1
- Number 2 is found at index 6
- Number 3 is found at index 9
- Number 4 is found at index 2
- Number 5 is found at index 4
- Number 6 is found at index 7
- Number 7 is found at index 13
- Number 8 is found at index 11
- Number 9 is found at index 5
$$E(1) = 90/10 = 9$$
I could probably compute $E(2)$ and $E(3)$ using a computer program.
What is an equation for $E(n)$? If too hard, what would be a good lower and upper bound for it? Has any work ever been done on this topic?