# Count ways to represent N as XOR of distinct integers not exceeding N

Given a positive integer **N**, the task is to find the number of ways to represent **N **as Bitwise XOR of distinct positive integers less than or equal to **N**.

**Examples:**

Input:N = 5Output:4Explanation:The given number N(= 5) can be represented as:

- 5 = 5
- 5 = (4 ^ 1)
- 5 = (5 ^ 3 ^ 2 ^ 1)
- 5 = (4 ^ 3 ^ 2)
Therefore, the total count is 4.

Input:N = 6Output:8

**Naive Approach:** The simplest approach to solve the problem is to find all subsets of first **N** natural numbers and count those subsets having Bitwise XOR value **N**. After checking for all the subsets, print the total value of the count obtained.

**Time Complexity:** O(N * 2^{N})**Auxiliary Space:** O(1)

**Efficient Approach:** The above approach can be optimized by using the observation that the number of ways to represent **N** as the **Bitwise XOR** of distinct positive integers is given by .

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include<bits/stdc++.h>` `using` `namespace` `std;` `// Function to count number of ways` `// to represent N as the Bitwise` `// XOR of distinct integers` `void` `countXorPartition(` `int` `N)` `{` ` ` ` ` `// Count number of subsets using` ` ` `// above-mentioned formula` ` ` `double` `a = ` `pow` `(2, ` `floor` `(N - ` `log` `(N + 1) /` ` ` `log` `(2)));` ` ` ` ` `// Print the resultant count` ` ` `cout << a;` `}` ` ` `// Driver Code` `int` `main()` `{` ` ` `int` `N = 5;` ` ` ` ` `countXorPartition(N);` `}` `// This code is contributed by SURENDRA_GANGWAR` |

## Java

`// java program for the above approach` `import` `java.io.*;` `class` `GFG{` `// Function to count number of ways` `// to represent N as the Bitwise` `// XOR of distinct integers` `static` `void` `countXorPartition(` `int` `N)` `{` ` ` ` ` `// Count number of subsets using` ` ` `// above-mentioned formula` ` ` `double` `a = Math.pow(` `2` `, (` `int` `)(N - Math.log(N + ` `1` `) /` ` ` `Math.log(` `2` `)));` ` ` ` ` `// Print the resultant count` ` ` `System.out.print(a);` `}` ` ` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `N = ` `5` `; ` ` ` `countXorPartition(N);` `}` `}` `// This code is contributed by shivanisinghss2110` |

## Python

`# Python program for the above approach` `from` `math ` `import` `*` `# Function to count number of ways` `# to represent N as the Bitwise` `# XOR of distinct integers` `def` `countXorPartition(N):` ` ` ` ` `# Count number of subsets using` ` ` `# above-mentioned formula` ` ` `a ` `=` `2` `*` `*` `floor(N ` `-` `log(N ` `+` `1` `)` `/` `log(` `2` `))` ` ` ` ` `# Print the resultant count` ` ` `print` `(` `int` `(a))` `# Driver Code` `N ` `=` `5` `countXorPartition(N)` |

## C#

`// C# program for the above approach` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG{` `// Function to count number of ways` `// to represent N as the Bitwise` `// XOR of distinct integers` `static` `void` `countXorPartition(` `int` `N)` `{` ` ` ` ` `// Count number of subsets using` ` ` `// above-mentioned formula` ` ` `double` `a = Math.Pow(2, (` `int` `)(N - Math.Log(N + 1) /` ` ` `Math.Log(2)));` ` ` ` ` `// Print the resultant count` ` ` `Console.Write(a);` `}` ` ` `// Driver Code` `public` `static` `void` `Main()` `{` ` ` `int` `N = 5; ` ` ` `countXorPartition(N);` `}` `}` `// This code is contributed by ipg2016107.` |

## Javascript

`<script>` `// JavaScript program for the above approach` `// Function to count number of ways` `// to represent N as the Bitwise` `// XOR of distinct integers` `function` `countXorPartition(N)` `{` ` ` ` ` `// Count number of subsets using` ` ` `// above-mentioned formula` ` ` `let a = Math.pow(2, Math.floor(N - Math.log(N + 1) /` ` ` `Math.log(2)));` ` ` ` ` `// Print the resultant count` ` ` `document.write(a);` `}` ` ` `// Driver Code` ` ` `let N = 5;` ` ` ` ` `countXorPartition(N);` `</script>` |

**Output:**

4

**Time Complexity:** O(1)**Auxiliary Space:** O(1)