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Count ways to express a number as sum of powers
• Difficulty Level : Medium
• Last Updated : 26 Mar, 2021

Given two integers x and n, we need to find number of ways to express x as sum of n-th powers of unique natural numbers. It is given that 1 <= n <= 20.
Examples:

```Input  : x = 100
n = 2
Output : 3
Explanation: There are three ways to
express 100 as sum of natural numbers
raised to power 2.
100 = 10^2 = 8^2+6^2 = 1^2+3^2+4^2+5^2+7^2

Input  : x = 100
n = 3
Output : 1
Explanation : The only combination is,
1^3 + 2^3 + 3^3 + 4^3```

We use recursion to solve the problem. We first check one by one that the number is included in summation or not.

## C++

 `// C++ program to count number of ways``// to express x as sum of n-th power``// of unique natural numbers.``#include ``using` `namespace` `std;` `// num is current num.``int` `countWaysUtil(``int` `x, ``int` `n, ``int` `num)``{``    ``// Base cases``    ``int` `val = (x - ``pow``(num, n));``    ``if` `(val == 0)``        ``return` `1;``    ``if` `(val < 0)``        ``return` `0;` `    ``// Consider two possibilities, num is``    ``// included and num is not included.``    ``return` `countWaysUtil(val, n, num + 1) +``           ``countWaysUtil(x, n, num + 1);``}` `// Returns number of ways to express``// x as sum of n-th power of two.``int` `countWays(``int` `x, ``int` `n)``{``    ``return` `countWaysUtil(x, n, 1);``}` `// Driver code``int` `main()``{``    ``int` `x = 100, n = 2;``    ``cout << countWays(x, n);``    ``return` `0;``}`

## Java

 `// Java program to count number of ways``// to express x as sum of n-th power``// of unique natural numbers.``public` `class` `GFG {` `    ``// num is current num.``    ``static` `int` `countWaysUtil(``int` `x, ``int` `n, ``int` `num)``    ``{``        ``// Base cases``        ``int` `val = (``int``) (x - Math.pow(num, n));``        ``if` `(val == ``0``)``            ``return` `1``;``        ``if` `(val < ``0``)``            ``return` `0``;``     ` `        ``// Consider two possibilities, num is``        ``// included and num is not included.``        ``return` `countWaysUtil(val, n, num + ``1``) +``               ``countWaysUtil(x, n, num + ``1``);``    ``}``     ` `    ``// Returns number of ways to express``    ``// x as sum of n-th power of two.``    ``static` `int` `countWays(``int` `x, ``int` `n)``    ``{``        ``return` `countWaysUtil(x, n, ``1``);``    ``}``     ` `    ``// Driver code``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `x = ``100``, n = ``2``;``        ``System.out.println(countWays(x, n));``    ``}``}``// This code is contributed by Sumit Ghosh`

## Python3

 `# Python program to count number of ways``# to express x as sum of n-th power``# of unique natural numbers.` `# num is current num.``def` `countWaysUtil(x,n,num):` `    ``# Base cases``    ``val ``=` `(x ``-` `pow``(num, n))``    ``if` `(val ``=``=` `0``):``        ``return` `1``    ``if` `(val < ``0``):``        ``return` `0`` ` `    ``# Consider two possibilities, num is``    ``# included and num is not included.``    ``return` `countWaysUtil(val, n, num ``+` `1``) ``+``\``           ``countWaysUtil(x, n, num ``+` `1``)` ` ` `# Returns number of ways to express``# x as sum of n-th power of two.``def` `countWays(x,n):``    ``return` `countWaysUtil(x, n, ``1``)` `    ` `# Driver code``x ``=` `100``n ``=` `2` `print``(countWays(x, n))` `# This code is contributed``# by Anant Agarwal.`

## C#

 `// C# program to count number of ways``// to express x as sum of n-th power``// of unique natural numbers.``using` `System;` `public` `class` `GFG {` `    ``// num is current num.``    ``static` `int` `countWaysUtil(``int` `x,``                          ``int` `n, ``int` `num)``    ``{``        ` `        ``// Base cases``        ``int` `val = (``int``) (x - Math.Pow(num, n));``        ``if` `(val == 0)``            ``return` `1;``        ``if` `(val < 0)``            ``return` `0;``    ` `        ``// Consider two possibilities,``        ``// num is included and num is``        ``// not included.``        ``return` `countWaysUtil(val, n, num + 1)``              ``+ countWaysUtil(x, n, num + 1);``    ``}``    ` `    ``// Returns number of ways to express``    ``// x as sum of n-th power of two.``    ``static` `int` `countWays(``int` `x, ``int` `n)``    ``{``        ``return` `countWaysUtil(x, n, 1);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `x = 100, n = 2;``        ` `        ``Console.WriteLine(countWays(x, n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

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## Javascript

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Output:

`3`

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