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 $x \leq m$.
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 the number of integers that meet the above criteria.
16 2 1000
2
Two: $400 = (4 + 0 + 0 + 16)^2$ and $841 = (8 + 4 + 1 + 16)^2$
0 3 5000
3
Three: $1=1^3$, $512 = (5 + 1 + 2)^3$ and $4913 = (4 + 9 + 1 + 3)^3$.
2 3 100000
0
There is no such number $x$ in the range below $100,000$ such that its sum of decimal digits $y$ satisfies the relation $(y+2)^3 = x$.