Generate elements of the array following given conditions

Given an integer N, for every integer i in the range 2 to N, assign a positive integer a_i such that the following conditions hold :

  • For any pair of indices (i, j), if i and j are coprime then a_i \neq a_j.
  • The maximum value of all a_i should be minimized (i.e. max value should be as small as possible).
  • A pair of integers is called coprime if their gcd(i, j) = 1.

    Examples:



    Input: N = 4
    Output: 1 2 1

    Input: N = 10
    Output: 1 2 1 3 2 4 1 2 3

    Approach: We have to keep two things in mind while assigning values at indices from 2 to n. First, for any pair (i, j), if i and j are coprime then we cannot assign same value to this pair of indices. Second, we have to minimize the maximum value assigned to each index from 2 to n.

    This can be achieved if distinct numbers are assigned to each prime, because all primes are coprime to each other. Then assign values at all composite positions same as any of its prime divisors. This solution works as for any pair (i, j), i is given the same number of a divisor and so is j, so if they are coprime, they cannot be given the same number.

    This approach can be implemented by making a small change when building the Sieve of Eratosthenes.
    When we met a prime for the first time, then assign a unique smallest value to it and all of its multiples. This way, all prime indices should have unique values and all composite indices have values same as any of its prime divisors.

    Below is the implementation of the above approach:

    C++

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    // C++ implementation of the approach
    #include <bits/stdc++.h>
    using namespace std;
      
    // Function to generate the required array
    void specialSieve(int n)
    {
      
        // Initialize cnt variable for assigning
        // unique value to prime and its multiples
        int cnt = 0;
        int prime[n + 1];
      
        for (int i = 0; i <= n; i++)
            prime[i] = 0;
      
        for (int i = 2; i <= n; i++) {
      
            // When we get a prime for the first time
            // then assign a unique smallest value to
            // it and all of its multiples
            if (!prime[i]) {
                cnt++;
                for (int j = i; j <= n; j += i)
                    prime[j] = cnt;
            }
        }
      
        // Print the generated array
        for (int i = 2; i <= n; i++)
            cout << prime[i] << " ";
    }
      
    // Driver code
    int main()
    {
        int n = 6;
      
        specialSieve(n);
      
        return 0;
    }

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    Java

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    // Java implementation of the approach
    import java.io.*;
      
    class GFG
    {
      
    // Function to generate the required array
    static void specialSieve(int n)
    {
      
        // Initialize cnt variable for assigning
        // unique value to prime and its multiples
        int cnt = 0;
        int prime[] = new int[n+1];
      
        for (int i = 0; i <= n; i++)
            prime[i] = 0;
      
        for (int i = 2; i <= n; i++)
        {
      
            // When we get a prime for the first time
            // then assign a unique smallest value to
            // it and all of its multiples
            if (!(prime[i]>0)) 
            {
                cnt++;
                for (int j = i; j <= n; j += i)
                    prime[j] = cnt;
            }
        }
      
        // Print the generated array
        for (int i = 2; i <= n; i++)
            System.out.print(prime[i] + " ");
    }
      
    // Driver code
    public static void main (String[] args)
    {
        int n = 6;
      
    specialSieve(n);
      
    }
    }
      
    // This code is contrubuted by anuj_67..

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    Python3

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    # Python3 implementation of the approach 
      
    # Function to generate the required array 
    def specialSieve(n) : 
      
        # Initialize cnt variable for assigning 
        # unique value to prime and its multiples 
        cnt = 0
        prime = [0]*(n + 1); 
      
        for i in range(2, n + 1) :
      
            # When we get a prime for the first time 
            # then assign a unique smallest value to 
            # it and all of its multiples 
            if (not prime[i]) :
                cnt += 1
                for j in range(i, n + 1, i) : 
                    prime[j] = cnt; 
      
        # Print the generated array 
        for i in range(2, n + 1) :
            print(prime[i],end = " "); 
      
      
    # Driver code 
    if __name__ == "__main__"
      
        n = 6
        specialSieve(n); 
          
    # This code is contributed by AnkitRai01

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    C#

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    // C# implementation of the approach
    using System;
      
    class GFG
    {
      
    // Function to generate the required array
    static void specialSieve(int n)
    {
      
        // Initialize cnt variable for assigning
        // unique value to prime and its multiples
        int cnt = 0;
        int []prime = new int[n+1];
      
        for (int i = 0; i <= n; i++)
            prime[i] = 0;
      
        for (int i = 2; i <= n; i++)
        {
      
            // When we get a prime for the first time
            // then assign a unique smallest value to
            // it and all of its multiples
            if (!(prime[i] > 0)) 
            {
                cnt++;
                for (int j = i; j <= n; j += i)
                    prime[j] = cnt;
            }
        }
      
        // Print the generated array
        for (int i = 2; i <= n; i++)
            Console.Write(prime[i] + " ");
    }
      
    // Driver code
    public static void Main ()
    {
        int n = 6;
      
        specialSieve(n);
      
    }
    }
      
    // This code is contrubuted by anuj_67..

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    Output:

    1 2 1 3 2
    

    Time Complexity: O(N Log N)



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    Improved By : vt_m, AnkitRai01