Problem:
Dudeney Number

　

A Dudeney number is a positive integer for which the sum of its decimal digits is equal to the cube root of the number. For example, $512$ is a Dudeney number because it is the cube of $8$, which is the sum of its decimal digits ($5 + 1 + 2$).

In this problem, we think of a type similar to Dudeney numbers and try to enumerate them.

Given a non-negative integer $a$, an integer $n$ greater than or equal to 2 and an upper limit $m$, make a program to enumerate all $x$’s such that the sum of its decimal digits $y$ satisfies the relation $x = (y + a)^n$, and $x \leq m$.

Input

The input is given in the following format.

$a$ $n$ $m$

The input line provides three integers: $a$ ($0 \leq a \leq 50$), $n$ ($2 \leq n \leq 10$) and the upper limit $m$ ($1000 \leq m \leq 10^8$).

Output

Output the number of integers that meet the above criteria.

Sample Input 1

16 2 1000

Sample Output 1

2

Two: $400 = (4 + 0 + 0 + 16)^2$ and $841 = (8 + 4 + 1 + 16)^2$

Sample Input 2

0 3 5000

Sample Output 2

3

Three: $1=1^3$, $512 = (5 + 1 + 2)^3$ and $4913 = (4 + 9 + 1 + 3)^3$.

Sample Input 3

2 3 100000

Sample Output 3

0

There is no such number $x$ in the range below $100,000$ such that its sum of decimal digits $y$ satisfies the relation $(y+2)^3 = x$.

Solution in Scala (judge verdict: Accepted)

import scala.annotation.tailrec

object Main extends App {
  import scala.io.StdIn._

  val Array(a, n, m) = readLine.trim.split(' ').map(_.toInt)
  println(
    count(a, n, m)
  )
  @tailrec def count(a: Int, n: Int, m: Int, y: Long = 1, res: Int = 0): Int = {
    val x = exponent(a + y, n)
    if (x > m) res
    else count(a, n, m, y + 1, if (sumDigits(x) == y) res + 1 else res)
  }
  def sumDigits(x: Long): Long = {
    x match {
      case 0L ⇒ 0
      case _ ⇒ x % 10 + sumDigits(x / 10)
    }
  }
  def exponent(a: Long, e: Int): Long = {
    e match {
      case 1 ⇒ a
      case _ ⇒ a * exponent(a, e - 1)
    }
  }
}