Mobius函数

莫比乌斯函数$\mu(n)$是组合函数中使用的乘法函数。它具有三个可能的值-1、0和1之一。
$For \ any \ positive \ integer \ n, \\ \(\mu(n) = \begin{cases} 1 & \text{ if } n=1, \\ 0 & \text{ if } a^2 \mid n \text{ for some } a > 1 \text{ (i.e., } n \text{ has a squared prime factor)}, \\ (-1)^k & \text { if } n \text{ is the product of } k \text{ distinct primes.} \ _\square \end{cases}\)$
例子：

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

方法1(简单)
我们遍历所有小于或等于N的数字i。对于每个数字，我们检查它是否除以N。如果是，则检查它是否也为质数。如果两个条件都满足，则检查其平方是否也除以N。如果是，则返回0。如果平方不除，则增加素数的计数。最后，如果素数个数为偶数，则返回1；如果素数个数为奇数，则返回-1。

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

Javascript

 
// JavaScript program for mobious 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/>");
             

输出：

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

方法2(高效)
这个想法是基于有效的程序来打印给定数量的所有素数。有趣的是，我们在这里不需要内部while循环，因为如果一个数字相除一次以上，我们可以立即返回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.

输出：

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

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