Program for Mobius Function

23 Jun, 2022
Mobius Function is a multiplicative function that is used in combinatorics. It has one of three possible values -1, 0 and 1.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++
Java
Python3
C#
PHP
Javascript
// 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 notbool 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 codeint 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 program for mobious functionimport 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
# 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 notdef 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 CodeN = 17print ("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# Program to evaluate Mobius Functionusing 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 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 notfunction 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.?>
<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++
Java
Python3
C#
PHP
Javascript
// 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 codeint 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 program to print all prime factorsimport 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.
# 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 notdef 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 CodeN = 17print ("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# program to print all prime factorsusing 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 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.?>
<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 
nitin mittal
vt_m
manishshaw1
souravghosh0416
mukesh07
anikakapoor
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