Split first N natural numbers into two subsequences with non-coprime sums

Given an integer N (N &e; 3), the task is to split all numbers from 1 to N into two subsequences such that the sum of two subsequences is non-coprime to each other.

Examples:

Input: N = 5
Output:
{1, 3, 5}
{2, 4}
Explanation: Sum of the subsequence X[] = 1 + 3 + 5 = 9.
Sum of the subsequence Y[] = 2 + 4 = 6.
Since GCD(9, 6) is 3, the sums are not co-prime to each other.

Input: N = 4
Output:
{1, 4}
{2, 3}

Naive Approach: The simplest approach is to split first N natural numbers into two subsequences in all possible ways and for each combination, check if the sum of both the subsequences is non-coprime or not. IF found to be true for any pair of subsequences, print that subsequence and break out of loop. 

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

Efficient Approach: The above approach can be optimized based on the following observations:

• The sum of the first (N – 1) natural numbers is given by (N*(N – 1))/2.
• Therefore, GCD of ((N*(N – 1))/2) and N is N.

From the above observation insert all the numbers from the range [1, N] in the one subsequence and N into another subsequence.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <iostream>
using namespace std;

// Function to split 1 to N into two subsequences with
// non-coprime sums
void printSubsequence(int N)
{
    cout << "{ ";
    for (int i = 1; i < N - 1; i++)
        cout << i << ", ";
    cout << N - 1 << " }\n";
    cout << "{ " << N << " }";
}
// Driver Code
int main()
{
    int N = 8;
    printSubsequence(N);
    return 0;
}

// This code is contributed by Sania Kumari Gupta

C

// C program for the above approach

#include <stdio.h>

// Function to split 1 to N into two subsequences with
// non-coprime sums
void printSubsequence(int N)
{
    printf("{ ");
    for (int i = 1; i < N - 1; i++)
        printf("%d ", i);
    printf("%d}\n", N - 1);
    printf("{%d}",N);
}
// Driver Code
int main()
{
    int N = 8;
    printSubsequence(N);
    return 0;
}

// This code is contributed by Sania Kumari Gupta

Java

// Java program for the above approach
import java.io.*;
public class GFG
{

  // Function to split 1 to N
  // into two subsequences
  // with non-coprime sums
  public static void printSubsequence(int N)
  {
    System.out.print("{ ");
    for (int i = 1; i < N - 1; i++)
    {
      System.out.print(i + ", ");
    }

    System.out.println(N - 1 + " }");
    System.out.print("{ " + N + " }");
  }

  // Driver code
  public static void main(String[] args)
  {
    int N = 8;
    printSubsequence(N);
  }
}

// This code is contributed by divyesh072019

Python3

# Python3 program for the above approach

# Function to split 1 to N
# into two subsequences
# with non-coprime sums
def printSubsequence(N):

    print("{ ", end = "")
    for i in range(1, N - 1):
        print(i, end = ", ")

    print(N - 1, end = " }\n")

    print("{", N, "}")

# Driver Code
if __name__ == '__main__':

    N = 8

    printSubsequence(N)

# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
class GFG
{

  // Function to split 1 to N
  // into two subsequences
  // with non-coprime sums
  public static void printSubsequence(int N)
  {
    Console.Write("{ ");
    for (int i = 1; i < N - 1; i++)
    {
        Console.Write(i + ", ");
    }

    Console.WriteLine(N - 1 + " }");
    Console.Write("{ " + N + " }");
  }

  // Driver code
  public static void Main(string[] args)
  {
    int N = 8;
    printSubsequence(N);
  }
}

// This code is contributed by AnkThon

Javascript

<script>

    // Javascript program for the above approach

    // Function to split 1 to N
    // into two subsequences
    // with non-coprime sums
    function printSubsequence(N)
    {
      document.write("{ ");
      for (let i = 1; i < N - 1; i++)
      {
          document.write(i + ", ");
      }

      document.write(N - 1 + " }" + "</br>");
      document.write("{ " + N + " }" + "</br>");
    }

    let N = 8;
    printSubsequence(N);

</script>

Output:

{ 1, 2, 3, 4, 5, 6, 7 }
{ 8 }

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