Generate Array such that max is minimized and arr[i] != arr[j] when j is a multiple of i

Given an integer N, the task is to generate an array arr[] having N positive integers such that arr[i] ≠ arr[j] if j is divisible by i (1-based indexing is considered) such that the maximum value of the sequence is minimum among all possible sequences.

Examples:

Input: N = 3
Output: 1 2 2
Explanation: In this sequence 2 is divisible by 1. arr[2] = 2, arr[1] = 1 and arr[2] ≠ arr[1].
In this sequence 3 is divisible by 1. arr[3] = 2, arr[1] = 1 and arr[3] ≠ arr[1].
The maximum value is 2 which is the minimum possible among all sequences.

Input: N = 1
Output: 1

Approach: This problem can be solved based on the following idea:

For each index i, the value of arr[i] must be at least same as the number of divisors it has (including the number itself). 
Instead of directly finding divisors for each number, one can also find multiples (say j) of each index i. and increment count of divisors of j.

Follow the illustration given below for a better understanding.

Say N = 3.
Create arr[] = {0, 0, 0} to store number of divisors of each value.

For i = 1:
            => Multiples of 1 are 1, 2 and 3.
            => Increment values of arr[1], arr[2] and arr[3].
            => So arr[] = {1, 1, 1}

For i = 1:
            => Multiples of 2 is 2.
            => Increment value of arr[2].
            => So arr[] = {1, 2, 1}

For i = 3:
            => Multiples of 3 is 3.
            => Increment value of arr[3].
            => So arr[] = {1, 2, 2}

So the final array is arr = {1, 2, 2} where the maximum value (i.e. 2) is minimum among all possible sequences.

Follow the steps mentioned below to implement the above observation:

• Create an N sized array (say arr[]) with all elements initially filled with 0.
• Use Sieve of Eratosthenes to find all multiples of all numbers from i = 1 to N.
• For this traverse from i = 1 to N:
  • Run a loop from j = i to N and increment j by i in each step:
    • Increment the value of arr[j] by 1.
• Return the final array.

Below is the implementation of the above approach.

C++

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to generate the array
vector<int> goodSequence(int N)
{
    vector<int> res(N);
    res[0] = 1;

    // Loop to implement
    // sieve of Eratosthenes
    for (int i = 1; i <= N; i++) {
        for (int j = 2 * i; j <= N; j += i)
            res[j - 1] = res[i - 1] + 1;
    }
    return res;
}

// Driver code
int main()
{
    int N = 3;

    // Function call
    vector<int> ans = goodSequence(N);
    for (int x : ans)
        cout << x << " ";
    return 0;
}

Java

// Java code to implement the approach
import java.io.*;

class GFG {

  // Function to generate the array
  static int[] goodSequence(int N)
  {
    int[] res = new int[N];
    res[0] = 1;

    // Loop to implement
    // sieve of Eratosthenes
    for (int i = 1; i <= N; i++) {
      for (int j = 2 * i; j <= N; j += i)
        res[j - 1] = res[i - 1] + 1;
    }
    return res;
  }

  // Driver code
  public static void main (String[] args) {
    int N = 3;

    // Function call
    int ans[] = goodSequence(N);
    for (int x = 0; x < ans.length; x++)
      System.out.print(ans[x] + " ");
  }

}

// This code is contributed by hrithikgarg03188.

Python3

# python3 code to implement the approach

import math

# Function to generate the array


def goodSequence(N):

    res = [0 for _ in range(N)]
    res[0] = 1

    # Loop to implement
    # sieve of Eratosthenes
    for i in range(1, N + 1):
        for j in range(2*i, N+1, i):
            res[j - 1] = res[i - 1] + 1

    return res


# Driver code
if __name__ == "__main__":

    N = 3

    # Function call
    ans = goodSequence(N)
    for x in ans:
        print(x, end=" ")

    # This code is contributed by rakeshsahni

C#

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

    // Function to generate the array
    static int[] goodSequence(int N)
    {
        int[] res = new int[N];
        res[0] = 1;

        // Loop to implement
        // sieve of Eratosthenes
        for (int i = 1; i <= N; i++) {
            for (int j = 2 * i; j <= N; j += i)
                res[j - 1] = res[i - 1] + 1;
        }
        return res;
    }

    // Driver code
    public static void Main()
    {
        int N = 3;

        // Function call
        int[] ans = goodSequence(N);
        for (int x = 0; x < ans.Length; x++)
            Console.Write(ans[x] + " ");
    }
}

// This code is contributed by Samim Hossain Mondal.

Javascript

<script>
       // JavaScript code for the above approach

       // Function to generate the array
       function goodSequence(N) {
           let res = new Array(N);
           res[0] = 1;

           // Loop to implement
           // sieve of Eratosthenes
           for (let i = 1; i <= N; i++) {
               for (let j = 2 * i; j <= N; j += i)
                   res[j - 1] = res[i - 1] + 1;
           }
           return res;
       }

       // Driver code
       let N = 3;

       // Function call
       let ans = goodSequence(N);
       for (let x of ans)
           document.write(x + " ")

       // This code is contributed by Potta Lokesh
   </script>

Output

1 2 2 

Time Complexity: O(N*log(log(N)))
Auxiliary Space: O(N)