Create Array of distinct elements where odd indexed elements are multiple of left neighbour

Given an integer N, the task is to generate an array A[] of length N such that it satisfies the following conditions for all 1 ? i ? N?1:

• A_i is multiple of A_i-1 when i is odd
• A_i is not multiple of A_i-1 when i is even
• All A_i are pairwise distinct
• 1 ? A_i ? 2?N

Note: If there are multiple answers print any of them.

Examples:

Input: N = 4
Output: 3 6 4 8 
?Explanation:  [3, 6, 4, 8] is a valid array because:
A1 = 3 is multiple of A2 = 6
A2 = 6 is not multiple of A3 = 4
A3 = 4 is multiple of A4 = 8.

Input: N = 6
Output: 4 8 5 10 6 12

Approach: The problem can be solved based on the following observation:

Observations:

Let x = N ? ?N / 2? + 1. Then, the following sequence is valid: [x, 2?x, x + 1, 2?(x + 1), x + 2, …]

• It is easy to see that the elements at odd indices are in increasing order from x?N. Similarly, the elements at even indices are in increasing order from 2?x?2?N (from 2?x?2?(N ? 1) when N is odd).
• Then, 2?x = 2?(N ? ?N / 2? + 1) > N implies the sets {x, x + 1, …, N} and {2?x, 2?(x + 1), …, 2?N} are disjoint. Therefore, all elements of the above sequence are unique.
• A_i is multiple of A_i-1 can be trivially verified to be true for all odd
  A_i is not multiple of A_i-1 holds true for all even i.

Thus, the provided sequence fulfills all requirements of the problem, and is therefore valid!

Follow the below steps to solve the problem:

• Initialize a variable oddElement = (N / 2) + 1 for odd indexed elements.
• Initialize a variable evenElement = oddElement * 2 for even indexed elements.
• Traverse a loop from 1 till N on i:
  • If i is odd print oddElement.
    • Assign evenElement = oddElement * 2.
    • Increment the evenElement.
  • Else print the evenElement.
    

Below is the implementation of the above approach.

C++

#include <iostream>
using namespace std;
#include <cmath>

// Function to find array
void find(int N)
{
    int oddElement = N - (int)floor(N / 2) + 1;
    int evenElement = 2 * oddElement;

    for (int i = 1; i <= N; i++) {

        // For odd indices
        if ((i % 2) != 0) {
            cout << oddElement << " ";
            evenElement = 2 * oddElement;
            oddElement++;
        }

        // For even indices
        else {
            cout << evenElement << " ";
        }
    }
}

// Driver Code
int main()
{
    int N = 4;

    // Function call
    find(N);
}

// This code is contributed by aarohirai2616.

Java

// Java code to implement the approach

import java.io.*;
import java.util.*;

public class GFG {

    // Function to find array
    public static void find(int N)
    {
        int oddElement = N - (int)Math.floor(N / 2) + 1;
        int evenElement = 2 * oddElement;

        for (int i = 1; i <= N; i++) {

            // For odd indices
            if ((i % 2) != 0) {
                System.out.print(oddElement + " ");
                evenElement = 2 * oddElement;
                oddElement++;
            }

            // For even indices
            else {
                System.out.print(evenElement + " ");
            }
        }
    }

    // Driver code
    public static void main(String[] args)
    {
        int N = 4;

        // Function call
        find(N);
    }
}

Python3

# Python code to implement the approach
import math

# Function to find array
def find(N):
    oddElement = int(N - (N/2) + 1)
    evenElement = 2 * oddElement

    for i in range(1, N + 1):

        # For odd indices
        if((i % 2) is not 0):
            print(oddElement, end=" ")
            evenElement = 2 * oddElement
            oddElement += 1

        # For even indices
        else:
            print(evenElement, end=" ")

N = 4

# Function call
find(N)

# This code is contributed by lokeshmvs21.

C#

// C# code to implement the approach
using System;
class GFG {


  // Function to find array
  public static void find(int N)
  {
    int oddElement = N - N / 2 + 1;
    int evenElement = 2 * oddElement;

    for (int i = 1; i <= N; i++) {

      // For odd indices
      if ((i % 2) != 0) {
        Console.Write(oddElement + " ");
        evenElement = 2 * oddElement;
        oddElement++;
      }

      // For even indices
      else {
        Console.Write(evenElement + " ");
      }
    }
  }


  // Driver code
  public static void Main(string[] args)
  {
    int N = 4;

    // Function call
    find(N);
  }
}

// This code is contributed by code_hunt.

Javascript

// Javascript  code to implement the approach

// Function to find array
function find(N){

    let oddElement = N - Math.floor(N / 2) + 1;
    let evenElement = 2 * oddElement;

    for(let i = 1; i <= N; i++) {

        // For odd indices
        if ((i % 2) != 0){
            process.stdout.write(oddElement + " ");
            evenElement = 2 * oddElement;
            oddElement++;
          }

        // For even indices
        else{
           process.stdout.write(evenElement + " ");
        }
    }
}

 // Driver Code
 let N = 4;

    // Function Call
 find(N);

 // This code is contributed by aarohirai2616.

Output

3 6 4 8 

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