## 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.

### To solve this problem, we will write a Python program to calculate the sum of all positive integers that do not have a zero in their digits and have a digital sum equal to a given number.

Here is the Python code to implement the function `f(n)`:

“`python

def f(n):

if n == 0:

return 0

if n == 1:

return 1

digits = [] # list to store the candidate digits

# Generate all possible digits that do not contain zero

for i in range(1, 10):

digits.append(i)

# Use dynamic programming to calculate f(n)

dp = [0] * (n + 1)

dp[0] = 1

for i in range(1, n + 1):

for digit in digits:

if i – digit >= 0:

dp[i] += dp[i – digit]

# Calculate the sum of all positive integers

# that do not have a zero in their digits and have a digital sum equal to n

total_sum = 0

for i in range(1, n + 1):

total_sum += dp[i] * i

return total_sum

“`

Now, we can use the above function to calculate the summation $\sum_{i=1}^{17} f(13^i)$:

“`python

total_sum = 0

for i in range(1, 18):

total_sum += f(13 ** i)

# Extract the last 9 digits from the total sum

last_nine_digits = total_sum % 1000000000

print(last_nine_digits)

“`

When we run the above code, it will output the last 9 digits of the sum $\sum_{i=1}^{17} f(13^i)$.

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