## Reflexive Position

### Problem 305

Published on Sunday, 10th October 2010, 04:00 am; Solved by 412; Difficulty rating: 60%
Let's call S the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) written down in base 10.

Thus, S = 1234567891011121314151617181920212223242...

It's easy to see that any number will show up an infinite number of times in S.

Let's call f(n) the starting position of the n^{th} occurrence of n in S.

For example, f(1)=1, f(5)=81, f(12)=271 and f(7780)=111111365.

Find ∑f(3^{k}) for 1≤k≤13.