梅森总理

Mersenne Prime是质数，小于1的2的幂。换句话说，如果任何素数的形式为2 ^k -1，其中k是大于或等于2的整数，则它是Mersenne素数。前几个Mersenne素数是3、7、31和127。
该任务将打印所有小于输入正整数n的Mersenne Prime。
例子：

Input: 10
Output: 3 7
3 and 7 are prime numbers smaller than or
equal to 10 and are of the form 2k-1

Input: 100
Output: 3 7 31 

想法是使用Eratosthenes筛子生成小于或等于给定数n的所有素数。生成所有此类素数后，我们将迭代2 ^k -1形式的所有数字，并检查它们是否为素数。
以下是该想法的实现。

C++

// Program to generate mersenne primes
#include
using namespace std;
 
// Generate all prime numbers less than n.
void SieveOfEratosthenes(int n, bool prime[])
{
    // Initialize all entries of boolean array
    // as true. A value in prime[i] will finally
    // be false if i is Not a prime, else true
    // bool prime[n+1];
    for (int i=0; i<=n; i++)
        prime[i] = true;
 
    for (int p=2; p*p<=n; p++)
    {
        // If prime[p] is not changed, then it
        // is a prime
        if (prime[p] == true)
        {
            // Update all multiples of p
            for (int i=p*2; i<=n; i += p)
                prime[i] = false;
        }
    }
}
 
// Function to generate mersenne primes less
// than or equal to n
void mersennePrimes(int n)
{
    // Create a boolean array "prime[0..n]"
    bool prime[n+1];
 
    // Generating primes using Sieve
    SieveOfEratosthenes(n,prime);
 
    // Generate all numbers of the form 2^k - 1
    // and smaller than or equal to n.
    for (int k=2; ((1<

---

Java

// Program to generate
// mersenne primes
import java.io.*;
 
class GFG {
     
    // Generate all prime numbers
    // less than n.
    static void SieveOfEratosthenes(int n,
                          boolean prime[])
    {
        // Initialize all entries of
        // boolean array as true. A
        // value in prime[i] will finally
        // be false if i is Not a prime,
        // else true bool prime[n+1];
        for (int i = 0; i <= n; i++)
            prime[i] = true;
     
        for (int p = 2; p * p <= n; p++)
        {
            // If prime[p] is not changed
            // , then it is a prime
            if (prime[p] == true)
            {
                // Update all multiples of p
                for (int i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
    }
     
    // Function to generate mersenne
    // primes lessthan or equal to n
    static void mersennePrimes(int n)
    {
        // Create a boolean array
        // "prime[0..n]"
        boolean prime[]=new boolean[n + 1];
     
        // Generating primes
        // using Sieve
        SieveOfEratosthenes(n, prime);
     
        // Generate all numbers of
        // the form 2^k - 1 and
        // smaller than or equal to n.
        for (int k = 2; (( 1 << k) - 1) <= n; k++)
        {
            long num = ( 1 << k) - 1;
     
            // Checking whether number is
            // prime and is one less then
            // the power of 2
            if (prime[(int)(num)])
                System.out.print(num + " ");
        }
    }
     
    // Driven program
    public static void main(String args[])
    {
        int n = 31;
        System.out.println("Mersenne prime"+
                     "numbers smaller than"+
                          "or equal to "+n);
         
        mersennePrimes(n);
    }
}
 
// This code is contributed by Nikita Tiwari.

Python3

# Program to generate mersenne primes
 
# Generate all prime numbers
# less than n.
def SieveOfEratosthenes(n, prime):
 
    # Initialize all entries of boolean
    # array as true. A value in prime[i]
    # will finally be false if i is Not
    # a prime, else true bool prime[n+1]
    for i in range(0, n + 1) :
        prime[i] = True
 
    p = 2
    while(p * p <= n):
     
        # If prime[p] is not changed,
        # then it is a prime
        if (prime[p] == True) :
         
            # Update all multiples of p
            for i in range(p * 2, n + 1, p):
                prime[i] = False
                 
        p += 1
         
# Function to generate mersenne
# primes less than or equal to n
def mersennePrimes(n) :
 
    # Create a boolean array
    # "prime[0..n]"
    prime = [0] * (n + 1)
 
    # Generating primes using Sieve
    SieveOfEratosthenes(n, prime)
 
    # Generate all numbers of the
    # form 2^k - 1 and smaller
    # than or equal to n.
    k = 2
    while(((1 << k) - 1) <= n ):
     
        num = (1 << k) - 1
 
        # Checking whether number
        # is prime and is one
        # less then the power of 2
        if (prime[num]) :
            print(num, end = " " )
             
        k += 1
     
# Driver Code
n = 31
print("Mersenne prime numbers smaller",
              "than or equal to " , n )
mersennePrimes(n)
 
# This code is contributed
# by Smitha

C#

// C# Program to generate mersenne primes
using System;
 
class GFG {
     
    // Generate all prime numbers less than n.
    static void SieveOfEratosthenes(int n,
                                bool []prime)
    {
         
        // Initialize all entries of
        // boolean array as true. A
        // value in prime[i] will finally
        // be false if i is Not a prime,
        // else true bool prime[n+1];
        for (int i = 0; i <= n; i++)
            prime[i] = true;
     
        for (int p = 2; p * p <= n; p++)
        {
             
            // If prime[p] is not changed,
            // then it is a prime
            if (prime[p] == true)
            {
                 
                // Update all multiples of p
                for (int i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
    }
     
    // Function to generate mersenne
    // primes lessthan or equal to n
    static void mersennePrimes(int n)
    {
         
        // Create a boolean array
        // "prime[0..n]"
        bool []prime = new bool[n + 1];
     
        // Generating primes
        // using Sieve
        SieveOfEratosthenes(n, prime);
     
        // Generate all numbers of
        // the form 2^k - 1 and
        // smaller than or equal to n.
        for (int k = 2; (( 1 << k) - 1) <= n; k++)
        {
            long num = ( 1 << k) - 1;
     
            // Checking whether number is
            // prime and is one less then
            // the power of 2
            if (prime[(int)(num)])
                Console.Write(num + " ");
        }
    }
     
    // Driven program
    public static void Main()
    {
        int n = 31;
         
        Console.WriteLine("Mersenne prime numbers"
               + " smaller than or equal to " + n);
         
        mersennePrimes(n);
    }
}
 
// This code is contributed by nitin mittal.

Javascript

 
// JavaScript to generate
// mersenne primes
 
    // Generate all prime numbers
    // less than n.
    function SieveOfEratosthenes( n,
                          prime)
    {
        // Initialize all entries of
        // boolean array as true. A
        // value in prime[i] will finally
        // be false if i is Not a prime,
        // else true bool prime[n+1];
        for (let i = 0; i <= n; i++)
            prime[i] = true;
       
        for (let p = 2; p * p <= n; p++)
        {
            // If prime[p] is not changed
            // , then it is a prime
            if (prime[p] == true)
            {
                // Update all multiples of p
                for (let i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
    }
       
    // Function to generate mersenne
    // primes lessthan or equal to n
    function mersennePrimes(n)
    {
        // Create a boolean array
        // "prime[0..n]"
        let prime=[];
       
        // Generating primes
        // using Sieve
        SieveOfEratosthenes(n, prime);
       
        // Generate all numbers of
        // the form 2^k - 1 and
        // smaller than or equal to n.
        for (let k = 2; (( 1 << k) - 1) <= n; k++)
        {
            let num = ( 1 << k) - 1;
       
            // Checking whether number is
            // prime and is one less then
            // the power of 2
            if (prime[(num)])
               document.write(num + " ");
        }
    }
 
// Driver Code
        let n = 31;
        document.write("Mersenne prime"+
                     "numbers smaller than"+
                          "or equal to "+n + "<br/>");
           
        mersennePrimes(n);
 
// This code is contributed by code_hunt.

输出：

Mersenne prime numbers smaller than or equal to 31
3 7 31