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Program for Mobius Function

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  • Difficulty Level : Medium
  • Last Updated : 23 Jun, 2022

Mobius Function \mu(n)        is a multiplicative function that is used in combinatorics. It has one of three possible values -1, 0 and 1.
For \ any \ positive \ integer \ n, \\ \(\mu(n) = \begin{cases} 1 & \text{ if } n=1, \\ 0 & \text{ if } a^2 \mid n \text{ for some } a > 1 \text{ (i.e., } n \text{ has a squared prime factor)}, \\ (-1)^k & \text { if } n \text{ is the product of } k \text{ distinct primes.} \ _\square \end{cases}\)
Examples: 
 

Input : 6
Output : 1
Solution: Prime Factors: 2 3.
Therefore p = 2, (-1)^p = 1

Input: 49
Output: 0
Solution: Prime Factors: 7 ( occurs twice). 
Since the prime factor occurs twice answer
is 0. 

Input: 3
Output: -1
Solution: Prime Factors: 3. Therefore p = 1, 
(-1) ^ p =-1

Input : 78
Output : 1
Solution: Prime Factors: 3, 13. Therefore p = 2, 
(-1)^p = 1

 

Method 1 (Simple) 
We iterate through all numbers i smaller than or equal to N. For every number we check if it divides N. If yes, we check if it’s also prime. If both conditions are satisfied, we check if its square also divides N. If yes, we return 0. If the square doesn’t divide, we increment count of prime factors. Finally, we return 1 if there are an even number of prime factors and return -1 if there are odd number of prime factors. 
 

C++




// CPP Program to evaluate Mobius Function
// M(N) = 1 if N = 1
// M(N) = 0 if any prime factor of N is contained twice
// M(N) = (-1)^(no of distinct prime factors)
#include<iostream>
using namespace std;
 
// Function to check if n is prime or not
bool isPrime(int n)
{
    if (n < 2)
        return false;
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0)
            return false;   
    return true;
}
 
int mobius(int N)
{
    // Base Case
    if (N == 1)
        return 1;
 
    // For a prime factor i check if i^2 is also
    // a factor.
    int p = 0;
    for (int i = 1; i <= N; i++) {
        if (N % i == 0 && isPrime(i)) {
 
            // Check if N is divisible by i^2
            if (N % (i * i) == 0)
                return 0;
            else
 
                // i occurs only once, increase f
                p++;
        }
    }
 
    // All prime factors are contained only once
    // Return 1 if p is even else -1
    return (p % 2 != 0)? -1 : 1;
}
 
// Driver code
int main()
{
    int N = 17;
    cout << "Mobius Functions M(N) at N = " << N << " is: "
         << mobius(N) << endl;
    cout << "Mobius Functions M(N) at N = " << 25 << " is: "
         << mobius(25) << endl;
    cout << "Mobius Functions M(N) at N = " << 6 << " is: "
         << mobius(6) << endl;
}

Java




// Java program for mobious function
import java.io.*;
public class GFG {
     
    // C# Program to evaluate Mobius
    // Function: M(N) = 1 if N = 1
    // M(N) = 0 if any prime factor
    // of N is contained twice
    // M(N) = (-1)^(no of distinct
    // prime factors)
 
    // Function to check if n is
    // prime or not
    static boolean isPrime(int n)
    {
        if (n < 2)
            return false;
        for (int i = 2; i * i <= n; i++)
            if (n % i == 0)
                return false;
        return true;
    }
 
    static int mobius(int N)
    {
        // Base Case
        if (N == 1)
            return 1;
 
        // For a prime factor i check if
        // i^2 is also a factor.
        int p = 0;
        for (int i = 1; i <= N; i++) {
            if (N % i == 0 && isPrime(i)) {
 
                // Check if N is divisible by i^2
                if (N % (i * i) == 0)
                    return 0;
                else
 
                    // i occurs only once, increase f
                    p++;
            }
        }
 
        // All prime factors are contained only
        // once Return 1 if p is even else -1
        return (p % 2 != 0) ? -1 : 1;
    }
 
    // Driver code
    static public void main(String[] args)
    {
        int N = 17;
        System.out.println("Mobius Functions M(N) at " +
                      " N = " + N + " is: "    + mobius(N));
        System.out.println("Mobius Functions M(N) at " +
                        " N = " + 25 + " is: " + mobius(25));
        System.out.println("Mobius Functions M(N) at " +
                          " N = " + 6 + " is: " + mobius(6));
    }
}
 
// This code is contributed by vt_m

Python3




# Python Program to
# evaluate Mobius def
# M(N) = 1 if N = 1
# M(N) = 0 if any
# prime factor of
# N is contained twice
# M(N) = (-1)^(no of
# distinct prime factors)
 
# def to check if
# n is prime or not
def isPrime(n) :
 
    if (n < 2) :
        return False
    for i in range(2, n + 1) :
        if (i * i <= n and n % i == 0) :
            return False
    return True
 
def mobius(N) :
     
    # Base Case
    if (N == 1) :
        return 1
 
    # For a prime factor i
    # check if i^2 is also
    # a factor.
    p = 0
    for i in range(1, N + 1) :
        if (N % i == 0 and
                isPrime(i)) :
 
            # Check if N is
            # divisible by i^2
            if (N % (i * i) == 0) :
                return 0
            else :
 
                # i occurs only once,
                # increase f
                p = p + 1
 
    # All prime factors are
    # contained only once
    # Return 1 if p is even
    # else -1
    if(p % 2 != 0) :
        return -1
    else :
        return 1
 
# Driver Code
N = 17
print ("Mobius defs M(N) at N = {} is: {}" .
         format(N, mobius(N)),end = "\n")
print ("Mobius defs M(N) at N = {} is: {}" .
        format(25, mobius(25)),end = "\n")
print ("Mobius defs M(N) at N = {} is: {}" .
          format(6, mobius(6)),end = "\n")
                                     
# This code is contributed by
# Manish Shaw(manishshaw1)

C#




// C# Program to evaluate Mobius Function
using System;
 
public class GFG
{
     
    // M(N) = 1 if N = 1
    // M(N) = 0 if any prime factor
    // of N is contained twice
    // M(N) = (-1)^(no of distinct
    // prime factors)
 
    // Function to check if n is
    // prime or not
    static bool isPrime(int n)
    {
        if (n == 2)
        return true;
 
        if (n % 2 == 0)
        return false;
        for (int i = 3; i * i <= n / 2; i += 2)
            if (n % i == 0)
            return false;
        return true;
    }
 
    static int mobius(int N)
    {
         
        // Base Case
        if (N == 1)
        return 1;
 
        // For a prime factor i check
        // if i^2 is also a factor.
        int p = 0;
        for (int i = 2; i <= N; i++)
        {
            if (N % i == 0 && isPrime(i)) {
 
                // Check if N is divisible by i^2
                if (N % (i * i) == 0)
                return 0;
                else
 
                // i occurs only once, increase f
                p++;
            }
        }
 
        // All prime factors are contained only
        // once Return 1 if p is even else -1
        return (p % 2 != 0) ? -1 : 1;
    }
 
    // Driver code
    static public void Main()
    {
         
         
        Console.WriteLine("Mobius Functions M(N) at " +
                         "N = " + 17 + " is: " + mobius(17));
        Console.WriteLine("Mobius Functions M(N) at " +
                         "N = " + 25 + " is: " + mobius(25));
        Console.WriteLine("Mobius Functions M(N) at " +
                          "N = " + 6 + " is: " + mobius(6));
         
    }
}
 
// This code is contributed by vt_m

PHP




<?php
// PHP Program to evaluate Mobius Function
// M(N) = 1 if N = 1
// M(N) = 0 if any prime factor of
// N is contained twice
// M(N) = (-1)^(no of distinct prime factors)
 
// Function to check if n is prime or not
function isPrime($n)
{
    if ($n < 2)
        return false;
    for ($i = 2; $i * $i <= $n; $i++)
        if ($n % $i == 0)
            return false;
    return true;
}
 
function mobius($N)
{
     
    // Base Case
    if ($N == 1)
        return 1;
 
    // For a prime factor i
    // check if i^2 is also
    // a factor.
    $p = 0;
    for ($i = 1; $i <= $N; $i++) {
        if ($N % $i == 0 && isPrime($i)) {
 
            // Check if N is divisible by i^2
            if ($N % ($i * $i) == 0)
                return 0;
            else
 
                // i occurs only once, increase f
                $p++;
        }
    }
 
    // All prime factors are
    // contained only once
    // Return 1 if p is even
    // else -1
    return ($p % 2 != 0) ? -1 : 1;
}
 
    // Driver Code
    $N = 17;
    echo "Mobius Functions M(N) at N = " ,$N , " is: "
                                  , mobius($N) ,"\n";
    echo "Mobius Functions M(N) at N = " ,25, " is: "
                                  , mobius(25),"\n" ;
    echo "Mobius Functions M(N) at N = " ,6, " is: "
                                       , mobius(6) ;
                                        
// This code is contributed by nitin mittal.
?>

Javascript




<script>
 
// JavaScript program for mobius function
 
  //  JavaScript Program to evaluate Mobius
    // Function: M(N) = 1 if N = 1
    // M(N) = 0 if any prime factor
    // of N is contained twice
    // M(N) = (-1)^(no of distinct
    // prime factors)
   
    // Function to check if n is
    // prime or not
    function isPrime(n)
    {
        if (n < 2)
            return false;
        for (let i = 2; i * i <= n; i++)
            if (n % i == 0)
                return false;
        return true;
    }
   
    function mobius(N)
    {
        // Base Case
        if (N == 1)
            return 1;
   
        // For a prime factor i check if
        // i^2 is also a factor.
        let p = 0;
        for (let i = 1; i <= N; i++) {
            if (N % i == 0 && isPrime(i)) {
   
                // Check if N is divisible by i^2
                if (N % (i * i) == 0)
                    return 0;
                else
   
                    // i occurs only once, increase f
                    p++;
            }
        }
   
        // All prime factors are contained only
        // once Return 1 if p is even else -1
        return (p % 2 != 0) ? -1 : 1;
    }
 
// Driver code   
          
        let N = 17;
        document.write("Mobius Functions M(N) at " +
                      " N = " + N + " is: "    + mobius(N) + "<br/>");
        document.write("Mobius Functions M(N) at " +
                        " N = " + 25 + " is: " + mobius(25) + "<br/>");
        document.write("Mobius Functions M(N) at " +
                          " N = " + 6 + " is: " + mobius(6) + "<br/>");
             
</script>

Output:  

Mobius Functions M(N) at N = 17 is: -1
Mobius Functions M(N) at N = 25 is: 0
Mobius Functions M(N) at N = 6 is: 1

Time Complexity: O(n√n ) 
Auxiliary Space: O(1)

Method 2 (Efficient) 
The idea is based on efficient program to print all prime factors of a given number. The interesting thing is, we do not need inner while loop here because if a number divides more than once, we can immediately return 0. 
 

C++




// Program to print all prime factors
# include <bits/stdc++.h>
using namespace std;
 
// Returns value of mobius()
int mobius(int n)
{
    int p = 0;
 
    // Handling 2 separately
    if (n%2 == 0)
    {
        n = n/2;
        p++;
 
        // If 2^2 also divides N
        if (n % 2 == 0)
           return 0;
    }
 
    // Check for all other prime factors
    for (int i = 3; i <= sqrt(n); i = i+2)
    {
        // If i divides n
        if (n%i == 0)
        {
            n = n/i;
            p++;
 
            // If i^2 also divides N
            if (n % i == 0)
               return 0;
        }
    }
 
    return (p % 2 == 0)? -1 : 1;
}
 
// Driver code
int main()
 {
    int N = 17;
    cout << "Mobius Functions M(N) at N = " << N << " is: "
         << mobius(N) << endl;
    cout << "Mobius Functions M(N) at N = " << 25 << " is: "
         << mobius(25) << endl;
    cout << "Mobius Functions M(N) at N = " << 6 << " is: "
         << mobius(6) << endl;
}

Java




// Java program to print all prime factors
import java.io.*;
 
class GFG {
     
    // Returns value of mobius()
    static int mobius(int n)
    {
        int p = 0;
     
        // Handling 2 separately
        if (n % 2 == 0)
        {
            n = n / 2;
            p++;
     
            // If 2^2 also divides N
            if (n % 2 == 0)
                return 0;
        }
     
        // Check for all other prime factors
        for (int i = 3; i <= Math.sqrt(n);
                                    i = i+2)
        {
            // If i divides n
            if (n % i == 0)
            {
                n = n / i;
                p++;
     
                // If i^2 also divides N
                if (n % i == 0)
                    return 0;
            }
        }
     
        return (p % 2 == 0)? -1 : 1;
    }
     
    // Driver code
    public static void main (String[] args)
    {
        int N = 17;
        System.out.println( "Mobius Functions"
               + " M(N) at N = " + N + " is: "
                                 + mobius(N));
        System.out.println ("Mobius Functions"
               + "M(N) at N = " + 25 + " is: "
                                + mobius(25));
        System.out.println( "Mobius Functions"
                + "M(N) at N = " + 6 + " is: "
                                 + mobius(6));
    }
}
 
// This code is contributed by anuj_67.

Python3




# Python Program to evaluate
# Mobius def M(N) = 1 if N = 1
# M(N) = 0 if any prime factor
# of N is contained twice
# M(N) = (-1)^(no of distinct
# prime factors)
import math
 
# def to check if n
# is prime or not
def isPrime(n) :
 
    if (n < 2) :
        return False
    for i in range(2, n + 1) :
        if (n % i == 0) :
            return False
        i = i * i
    return True
 
def mobius(n) :
 
    p = 0
 
    # Handling 2 separately
    if (n % 2 == 0) :
     
        n = int(n / 2)
        p = p + 1
 
        # If 2^2 also
        # divides N
        if (n % 2 == 0) :
            return 0
     
 
    # Check for all
    # other prime factors
    for i in range(3, int(math.sqrt(n)) + 1) :
     
        # If i divides n
        if (n % i == 0) :
         
            n = int(n / i)
            p = p + 1
 
            # If i^2 also
            # divides N
            if (n % i == 0) :
                return 0
        i = i + 2   
     
    if(p % 2 == 0) :
        return -1
    else :
        return 1
 
# Driver Code
N = 17
print ("Mobius defs M(N) at N = {} is: {}\n" .
                        format(N, mobius(N)));
print ("Mobius defs M(N) at N = 25 is: {}\n" .
                          format(mobius(25)));
print ("Mobius defs M(N) at N = 6 is: {}\n" .
                          format(mobius(6)));
                                         
# This code is contributed by
# Manish Shaw(manishshaw1)

C#




// C# program to print all prime factors
using System;
class GFG {
     
    // Returns value of mobius()
    static int mobius(int n)
    {
        int p = 0;
     
        // Handling 2 separately
        if (n % 2 == 0)
        {
            n = n / 2;
            p++;
     
            // If 2^2 also divides N
            if (n % 2 == 0)
                return 0;
        }
     
        // Check for all other prime factors
        for (int i = 3; i <= Math.Sqrt(n);
                                    i = i+2)
        {
            // If i divides n
            if (n % i == 0)
            {
                n = n / i;
                p++;
     
                // If i^2 also divides N
                if (n % i == 0)
                    return 0;
            }
        }
     
        return (p % 2 == 0)? -1 : 1;
    }
     
    // Driver Code
    public static void Main ()
    {
        int N = 17;
        Console.WriteLine( "Mobius Functions"
              + " M(N) at N = " + N + " is: "
                                + mobius(N));
        Console.WriteLine("Mobius Functions"
             + "M(N) at N = " + 25 + " is: "
                                + mobius(25));
        Console.WriteLine( "Mobius Functions"
               + "M(N) at N = " + 6 + " is: "
                                + mobius(6));
    }
}
 
// This code is contributed by anuj_67.

PHP




<?php
// PHP Program to print
// all prime factors
 
// Returns value of mobius()
function mobius( $n)
{
    $p = 0;
 
    // Handling 2 separately
    if ($n % 2 == 0)
    {
        $n = $n / 2;
        $p++;
 
        // If 2^2 also divides N
        if ($n % 2 == 0)
        return 0;
    }
 
    // Check for all
    // other prime factors
    for ( $i = 3; $i <= sqrt($n); $i = $i + 2)
    {
         
        // If i divides n
        if ($n % $i == 0)
        {
            $n = $n / $i;
            $p++;
 
            // If i^2 also divides N
            if ($n % $i == 0)
            return 0;
        }
    }
 
    return ($p % 2 == 0)? -1 : 1;
}
 
    // Driver code
    $N = 17;
    echo "Mobius Functions M(N) at N = ", $N, " is: "
        , mobius($N),"\n" ;
    echo "Mobius Functions M(N) at N = " , 25 , " is: "
        , mobius(25),"\n";
    echo "Mobius Functions M(N) at N = " , 6 , " is: "
        , mobius(6) ;
 
// This code is contributed by anuj_67.
?>

Javascript




<script>
 
    // JavaScript program to print all prime factors
     
    // Returns value of mobius()
    function mobius(n)
    {
        let p = 0;
      
        // Handling 2 separately
        if (n % 2 == 0)
        {
            n = parseInt(n / 2, 10);
            p++;
      
            // If 2^2 also divides N
            if (n % 2 == 0)
                return 0;
        }
      
        // Check for all other prime factors
        for (let i = 3; i <= Math.sqrt(n); i = i+2)
        {
            // If i divides n
            if (n % i == 0)
            {
                n = parseInt(n / i, 10);
                p++;
      
                // If i^2 also divides N
                if (n % i == 0)
                    return 0;
            }
        }
      
        return (p % 2 == 0)? -1 : 1;
    }
     
    let N = 17;
    document.write( "Mobius Functions"
                      + " M(N) at N = " + N + " is: "
                      + mobius(N) + "</br>");
    document.write("Mobius Functions"
                      + "M(N) at N = " + 25 + " is: "
                      + mobius(25) + "</br>");
    document.write( "Mobius Functions"
                      + "M(N) at N = " + 6 + " is: "
                      + mobius(6));
     
</script>

Output: 

Mobius Functions M(N) at N = 17 is: -1
Mobius Functions M(N) at N = 25 is: 0
Mobius Functions M(N) at N = 6 is: 1

Time Complexity: O(√n) 
Auxiliary Space: O(1)

Please suggest if someone has a better solution which is more efficient in terms of space and time.
This article is contributed by Aarti_Rathi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

References 
1) http://mathworld.wolfram.com/MobiusFunction.html 
2) https://en.wikipedia.org/wiki/M%C3%B6bius_function 
3) https://en.wikipedia.org/wiki/Completely_multiplicative_function
 


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