# Program for Mobius Function

Mobius Function is a multiplicative function which 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


## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 square doesn’t divide, we increment count of prime factors. Finally we return 1 if there are 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  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

 

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


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  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 prine 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 prine 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 prine 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 prine 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

 

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


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