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Sum of digits, experience 13

Problem 377

Published on Sunday, 25th March 2012, 05:00 am; Solved by 385

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.