Given a positive integer n, let S(n) denote the digit sum of n. Consider the sequence of numbers given by
n₁ = S(n)
nk = S(nk−1)k≥2
For how many positive integers n ≤ 2013 does the sequence {nk} contain the number 9?
To find the number of positive integers n ≤ 2013 for which the sequence {nk} contains the number 9, we need to calculate the sequence for each n and check if 9 appears in it.
Here are the steps to find the answer:
1. Define a function S(n) that calculates the digit sum of n. You can do this by converting the number n to a string, iterating through each digit, and summing them up.
2. Initialize a counter variable, count, to keep track of the number of integers n that satisfy the condition.
3. Iterate through each positive integer n from 1 to 2013.
4. For each n, calculate the sequence {nk} using the formula nk = S(nk−1), starting with k = 1 and continuing until nk contains the number 9 or exceeds 2013.
5. Within the sequence calculation loop, check if 9 appears in the current value of nk. If it does, increment the count variable and break out of the loop.
6. After the iteration is complete, the value of count will represent the number of positive integers n ≤ 2013 for which the sequence {nk} contains the number 9.
Implementing this algorithm will give you the desired count.