yaworsw/euler-manager

---
:id: 414
:name: Kaprekar constant
:url: https://projecteuler.net/problem=414
:content: "6174 is a remarkable number; if we sort its digits in increasing order
  and subtract that number from the number you get when you sort the digits in decreasing
  order, we get 7641-1467=6174.  \nEven more remarkable is that if we start from any
  4 digit number and repeat this process of sorting and subtracting, we'll eventually
  end up with 6174 or immediately with 0 if all digits are equal.  \n This also works
  with numbers that have less than 4 digits if we pad the number with leading zeroes
  until we have 4 digits.  \nE.g. let's start with the number 0837:  \n8730-0378=8352
  \ \n8532-2358=6174\n\n6174 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**.\n\nWe can consider the Kaprekar routine for
  other bases and number of digits.  \n 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.  \nHowever, it can be shown that for 5 digits and a base b = 6t+3≠9,
  a Kaprekar constant exists.  \nE.g. base 15: (10,4,14,9,5)<sub>15</sub>  \nbase
  21: (14,6,20,13,7)<sub>21</sub>\n\nDefine <var>C<sub>b</sub></var> to be the Kaprekar
  constant in base <var>b</var> for 5 digits. Define the function <var>sb(i)</var>
  to be\n\n- 0 if i = <var>C<sub>b</sub></var> or if <var>i</var> written in base
  <var>b</var> consists of 5 identical digits\n- the number of iterations it takes
  the Kaprekar routine in base <var>b</var> to arrive at <var>C<sub>b</sub></var>,
  otherwise\nNote that we can define <var>sb(i)</var> for all integers <var>i</var>
  \\< <var>b</var><sup>5</sup>. If <var>i</var> written in base <var>b</var> takes
  less than 5 digits, the number is padded with leading zero digits until we have
  5 digits before applying the Kaprekar routine.\n\nDefine <var>S(b)</var> as the
  sum of <var>sb(i)</var> for 0 \\< <var>i</var> \\< <var>b</var><sup>5</sup>.  \nE.g.
  S(15) = 5274369  \n S(111) = 400668930299\n\nFind the sum of S(6k+3) for 2 ≤ k ≤
  300.  \nGive the last 18 digits as your answer.\n\n"