Mobius函数|套装2

给定整数N。任务是找到从1到N的所有数字的Mobius函数。

例子：

Input: N = 5
Output: 1 -1 -1 0 -1

Input: N = 10
Output: 1 -1 -1 0 -1 1 -1 0 0 1

方法：想法是首先使用Eratosthenes筛子找到从1到N的所有数字中的最小素数，然后使用这些最小素数，可以为所有数字计算Mobius函数，具体取决于一个数字是否包含奇数个不同的素数或不同数量的素数。

下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
using namespace std;
#define N 100005
  
int lpf[N];
  
// Function to calculate least
// prime factor of each number
void least_prime_factor()
{
    for (int i = 2; i < N; i++)
  
        // If it is a prime number
        if (!lpf[i])
  
            for (int j = i; j < N; j += i)
  
                // For all multiples which are not
                // visited yet.
                if (!lpf[j])
                    lpf[j] = i;
}
  
// Function to find the value of Mobius function
// for all the numbers from 1 to n
void Mobius(int n)
{
    // To store the values of Mobius function
    int mobius[N];
  
    for (int i = 1; i < N; i++) {
  
        // If number is one
        if (i == 1)
            mobius[i] = 1;
        else {
  
            // If number has a squared prime factor
            if (lpf[i / lpf[i]] == lpf[i])
                mobius[i] = 0;
  
            // Multiply -1 with the previous number
            else
                mobius[i] = -1 * mobius[i / lpf[i]];
        }
    }
  
    for (int i = 1; i <= n; i++)
        cout << mobius[i] << " ";
}
  
// Driver code
int main()
{
    int n = 5;
  
    // Function to find least prime factor
    least_prime_factor();
  
    // Function to find mobius function
    Mobius(n);
}

Java

// Java implementation of the approach
import java.util.*;
  
class GFG 
{
  
static int N = 100005;
  
static int []lpf = new int[N];
  
// Function to calculate least
// prime factor of each number
static void least_prime_factor()
{
    for (int i = 2; i < N; i++)
  
        // If it is a prime number
        if (lpf[i] % 2 != 1)
  
            for (int j = i; j < N; j += i)
  
                // For all multiples which are not
                // visited yet.
                if (lpf[j] % 2 != 0)
                    lpf[j] = i;
}
  
// Function to find the value of Mobius function
// for all the numbers from 1 to n
static void Mobius(int n)
{
    // To store the values of Mobius function
    int []mobius = new int[N];
  
    for (int i = 1; i < N; i++) 
    {
  
        // If number is one
        if (i == 1)
            mobius[i] = 1;
        else
        {
  
            // If number has a squared prime factor
            if (lpf[i / lpf[i]] == lpf[i])
                mobius[i] = 0;
  
            // Multiply -1 with the previous number
            else
                mobius[i] = -1 * mobius[i / lpf[i]];
        }
    }
  
    for (int i = 1; i <= n; i++)
        System.out.print(mobius[i] + " ");
}
  
// Driver code
public static void main(String[] args)
{
    int n = 5;
    Arrays.fill(lpf, -1);
      
    // Function to find least prime factor
    least_prime_factor();
  
    // Function to find mobius function
    Mobius(n);
}
} 
  
// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of the approach 
N = 100005
  
lpf = [0] * N; 
  
# Function to calculate least 
# prime factor of each number 
def least_prime_factor() :
  
    for i in range(2, N) :
  
        # If it is a prime number 
        if (not lpf[i]) :
  
            for j in range(i, N, i) : 
  
                # For all multiples which are not 
                # visited yet. 
                if (not lpf[j]) :
                    lpf[j] = i; 
  
# Function to find the value of Mobius function 
# for all the numbers from 1 to n 
def Mobius(n) :
  
    # To store the values of Mobius function 
    mobius = [0] * N; 
  
    for i in range(1, N) :
  
        # If number is one 
        if (i == 1) :
            mobius[i] = 1; 
        else :
  
            # If number has a squared prime factor 
            if (lpf[i // lpf[i]] == lpf[i]) :
                mobius[i] = 0; 
  
            # Multiply -1 with the previous number 
            else :
                mobius[i] = -1 * mobius[i // lpf[i]]; 
  
    for i in range(1, n + 1) :
        print(mobius[i], end = " "); 
  
# Driver code 
if __name__ == "__main__" : 
  
    n = 5; 
  
    # Function to find least prime factor 
    least_prime_factor(); 
  
    # Function to find mobius function 
    Mobius(n); 
  
# This code is contributed by AnkitRai01

C#

// C# implementation of the approach
using System;
  
class GFG
{
static int N = 100005;
  
static int []lpf = new int[N];
  
// Function to calculate least
// prime factor of each number
static void least_prime_factor()
{
    for (int i = 2; i < N; i++)
  
        // If it is a prime number
        if (lpf[i] % 2 != 1)
  
            for (int j = i; j < N; j += i)
  
                // For all multiples which 
                // are not visited yet.
                if (lpf[j] % 2 != 0)
                    lpf[j] = i;
}
  
// Function to find the value of 
// Mobius function for all the numbers
// from 1 to n
static void Mobius(int n)
{
    // To store the values of 
    // Mobius function
    int []mobius = new int[N];
  
    for (int i = 1; i < N; i++) 
    {
  
        // If number is one
        if (i == 1)
            mobius[i] = 1;
        else
        {
  
            // If number has a squared prime factor
            if (lpf[i / lpf[i]] == lpf[i])
                mobius[i] = 0;
  
            // Multiply -1 with the 
            // previous number
            else
                mobius[i] = -1 * mobius[i / lpf[i]];
        }
    }
  
    for (int i = 1; i <= n; i++)
        Console.Write(mobius[i] + " ");
}
  
// Driver code
static public void Main ()
{
    int n = 5;
    Array.Fill(lpf, -1);
      
    // Function to find least prime factor
    least_prime_factor();
      
    // Function to find mobius function
    Mobius(n);
}
} 
  
// This code is contributed by ajit.

输出：

1 -1 -1 0 -1