## Sum of digits, experience 13

### Problem 377

There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely:

5, 14, 23, 32, 41, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111 and 11111.

Their sum is 17891.

Let `f`(`n`) be the sum of all positive integers that do not have a zero in their digits and have a digital sum equal to `n`.

Find $\displaystyle \sum_{i=1}^{17} f(13^i)$.

Give the last 9 digits as your answer.