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Problem 377

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Sum of digits, experience 13

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 $\sum_{i=1}^{17}f(13^i)$.
Give the last 9 digits as your answer.

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数字和，邂逅13

数字中不包含零且各位数字之和等于5的正整数一共有16个，分别是：
5、14、23、32、41、113、122、131、212、221、311、1112、1121、1211、2111和11111。
它们的和是17891。

数字中不包含零且各位数字之和等于n的正整数之和记为f(n)。

求$\sum_{i=1}^{17}f(13^i)$。
给出其最后9位数字作为你的答案。

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