2.5 KiB
id | challengeType | title |
---|---|---|
5900f50b1000cf542c51001d | 5 | Problem 414: Kaprekar constant |
Description
6174 is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the Kaprekar routine.
We can consider the Kaprekar routine for other bases and number of digits. Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers. However, it can be shown that for 5 digits and a base b = 6t+3≠9, a Kaprekar constant exists. E.g. base 15: (10,4,14,9,5)15 base 21: (14,6,20,13,7)21
Define Cb to be the Kaprekar constant in base b for 5 digits. Define the function sb(i) to be 0 if i = Cb or if i written in base b consists of 5 identical digits the number of iterations it takes the Kaprekar routine in base b to arrive at Cb, otherwise
Note that we can define sb(i) for all integers i < b5. If i written in base b takes less than 5 digits, the number is padded with leading zero digits until we have 5 digits before applying the Kaprekar routine.
Define S(b) as the sum of sb(i) for 0 < i < b5. E.g. S(15) = 5274369 S(111) = 400668930299
Find the sum of S(6k+3) for 2 ≤ k ≤ 300. Give the last 18 digits as your answer.
Instructions
Tests
tests:
- text: <code>euler414()</code> should return 552506775824935500.
testString: assert.strictEqual(euler414(), 552506775824935500, '<code>euler414()</code> should return 552506775824935500.');
Challenge Seed
function euler414() {
// Good luck!
return true;
}
euler414();
Solution
// solution required