[L, R] 中 N 的所有值的计数，使得直到 N 的素数计数也是素数

给定两个正整数L和R ，任务是找出范围[L, R]之间的值总数，使得从1到N的素数计数也是素数。
例子：

Input: L = 3, R = 10 
Output: 4 
Explanation: 
Number of primes upto 3, 4, 5, 6, 7, 8, 9 and 10 are 2, 2, 3, 3, 4, 4, 4 and 4 respectively. So, there are a total 4 such numbers {3, 4, 5, 6}[3, 10].
Input: L = 4, R = 12 
Output: 5 
Explanation: 
Number of primes upto 4, 5, 6, 7, 8, 9, 10, 11 and 12 are 2, 3, 3, 4, 4, 4, 4, 5 and 5 respectively. So, there are total 5 such numbers {4, 5, 6, 11, 12} which satisfy the condition in the range [4, 12]. 
 

天真的方法：
解决该问题的最简单的方法是遍历用于在范围[1，L – 1]的所有值计算在该范围内的素数的数量。计算一次，检查计数是否为素数。现在，开始一一遍历[L, R]范围内的值并检查数字是否为素数并相应地增加计数。对于每个更新的计数，检查它是否是质数，并相应地更新给定范围内所需数字的计数。
时间复杂度： O(R ² )
有效的方法：
上述方法可以通过埃拉托色尼筛法进一步优化。请按照以下步骤解决问题：

1. 使用筛子找出直到R 的所有素数。
2. 维护一个频率数组，以存储直到R 的所有值的素数计数。
3. 创建另一个计数数组(比如freqPrime[] )，如果到i的总素数的累积计数本身就是一个素数，则在位置i处加 1。
4. 现在对于任何范围L 到 R ，那么疯狂素数的数量可以通过freqPrime[R] – freqPrime[L – 1]来计算。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to count the number of
// crazy primes in the given range [L, R]
int count_crazy_primes(int L, int R)
{
    // Stores all primes
    int prime[R + 1] = { 0 };
 
    // Stores count of primes
    int countPrime[R + 1] = { 0 };
 
    // Stores if frequency of
    // primes is a prime or not
    // upto each index
    int freqPrime[R + 1] = { 0 };
 
    prime[0] = prime[1] = 1;
    // Sieve of Eratosthenes
    for (int p = 2; p * p <= R; p++) {
        if (prime[p] == 0) {
            for (int i = p * p;
                 i <= R; i += p)
 
                prime[i] = 1;
        }
    }
 
    // Count primes
    for (int i = 1; i <= R; i++) {
        countPrime[i] = countPrime[i - 1];
 
        // If i is a prime
        if (!prime[i]) {
            countPrime[i]++;
        }
    }
 
    // Stores frequency of primes
    for (int i = 1; i <= R; i++) {
        freqPrime[i] = freqPrime[i - 1];
 
        // If the frequency of primes
        // is a prime
        if (!prime[countPrime[i]]) {
 
            // Increase count of
            // required numbers
            freqPrime[i]++;
        }
    }
 
    // Return the required count
    return (freqPrime[R]
            - freqPrime[L - 1]);
}
 
// Driver Code
int main()
{
    // Given Range
    int L = 4, R = 12;
 
    // Function Call
    cout << count_crazy_primes(L, R);
    return 0;
}

Java

// Java implementation of the approach
class GFG{
 
// Function to count the number of
// crazy primes in the given range [L, R]
static int count_crazy_primes(int L, int R)
{
     
    // Stores all primes
    int prime[] = new int[R + 1];
 
    // Stores count of primes
    int countPrime[] = new int[R + 1];
 
    // Stores if frequency of
    // primes is a prime or not
    // upto each index
    int freqPrime[] = new int[R + 1];
     
    prime[0] = 1;
    prime[1] = 1;
     
    // Sieve of Eratosthenes
    for(int p = 2; p * p <= R; p++)
    {
       if (prime[p] == 0)
       {
           for(int i = p * p;
                   i <= R; i += p)
              prime[i] = 1;
       }
    }
 
    // Count primes
    for(int i = 1; i <= R; i++)
    {
       countPrime[i] = countPrime[i - 1];
        
       // If i is a prime
       if (prime[i] != 0)
       {
           countPrime[i]++;
       }
    }
 
    // Stores frequency of primes
    for(int i = 1; i <= R; i++)
    {
       freqPrime[i] = freqPrime[i - 1];
        
       // If the frequency of primes
       // is a prime
       if (prime[countPrime[i]] != 0)
       {
            
           // Increase count of
           // required numbers
           freqPrime[i]++;
       }
    }
 
    // Return the required count
    return (freqPrime[R] -
            freqPrime[L - 1]);
}
 
// Driver code
public static void main (String[] args)
{
     
    // Given range
    int L = 4, R = 12;
 
    // Function call
    System.out.println(count_crazy_primes(L, R));
}
}
 
// This code is contributed by Pratima Pandey

Python3

# Python3 program for the above approach
 
# Function to count the number of
# crazy primes in the given range [L, R]
def count_crazy_primes(L, R):
 
    # Stores all primes
    prime = [0] * (R + 1)
 
    # Stores count of primes
    countPrime = [0] * (R + 1)
 
    # Stores if frequency of
    # primes is a prime or not
    # upto each index
    freqPrime = [0] * (R + 1)
 
    prime[0] = prime[1] = 1
     
    # Sieve of Eratosthenes
    p = 2
    while p * p <= R:
        if (prime[p] == 0):
            for i in range (p * p,
                            R + 1 , p):
                prime[i] = 1
        p += 1
    
    # Count primes
    for i in range (1 , R + 1):
        countPrime[i] = countPrime[i - 1]
 
        # If i is a prime
        if (not prime[i]):
            countPrime[i] += 1
 
    # Stores frequency of primes
    for i in range (1, R + 1):
        freqPrime[i] = freqPrime[i - 1]
 
        # If the frequency of primes
        # is a prime
        if (not prime[countPrime[i]]):
 
            # Increase count of
            # required numbers
            freqPrime[i] += 1
 
    # Return the required count
    return (freqPrime[R] -
            freqPrime[L - 1])
 
# Driver Code
if __name__ =="__main__":
   
    # Given Range
    L = 4
    R = 12
 
    # Function Call
    print(count_crazy_primes(L, R))
 
# This code is contributed by Chitranayal

C#

// C# implementation of the approach
using System;
class GFG{
   
// Function to count the number of
// crazy primes in the given range [L, R]
static int count_crazy_primes(int L,
                              int R)
{     
    // Stores all primes
    int []prime = new int[R + 1];
   
    // Stores count of primes
    int []countPrime = new int[R + 1];
   
    // Stores if frequency of
    // primes is a prime or not
    // upto each index
    int []freqPrime = new int[R + 1];
       
    prime[0] = 1;
    prime[1] = 1;
       
    // Sieve of Eratosthenes
    for(int p = 2; p * p <= R; p++)
    {
       if (prime[p] == 0)
       {
           for(int i = p * p; i <= R;
               i += p)
              prime[i] = 1;
       }
    }
   
    // Count primes
    for(int i = 1; i <= R; i++)
    {
       countPrime[i] = countPrime[i - 1];
          
       // If i is a prime
       if (prime[i] != 0)
       {
           countPrime[i]++;
       }
    }
   
    // Stores frequency of primes
    for(int i = 1; i <= R; i++)
    {
       freqPrime[i] = freqPrime[i - 1];
          
       // If the frequency of primes
       // is a prime
       if (prime[countPrime[i]] != 0)
       {            
           // Increase count of
           // required numbers
           freqPrime[i]++;
       }
    }
   
    // Return the required count
    return (freqPrime[R] -
            freqPrime[L - 1]);
}
   
// Driver code
public static void Main(String[] args)
{      
    // Given range
    int L = 4, R = 12;
   
    // Function call
    Console.WriteLine(
            count_crazy_primes(L, R));
}
}
// This code is contributed by 29AjayKumar

Javascript

// Javascript program for the above approach
 
// Function to count the number of
// crazy primes in the given range [L, R]
function count_crazy_primes(L, R)
{
     
    // Stores all primes
    let prime = Array.from({length: R+1}, (_, i) => 0);
 
    // Stores count of primes
    let countPrime = Array.from({length: R+1}, (_, i) => -1);
 
    // Stores if frequency of
    // primes is a prime or not
    // upto each index
    let freqPrime = Array.from({length: R+1}, (_, i) => 0);
     
    prime[0] = 1;
    prime[1] = 1;
     
    // Sieve of Eratosthenes
    for(let p = 2; p * p <= R; p++)
    {
       if (prime[p] == 0)
       {
           for(let i = p * p;
                   i <= R; i += p)
              prime[i] = 1;
       }
    }
 
    // Count primes
    for(let i = 1; i <= R; i++)
    {
       countPrime[i] = countPrime[i - 1];
        
       // If i is a prime
       if (prime[i] != 0)
       {
           countPrime[i]++;
       }
    }
    // Stores frequency of primes
    for (let i = 1; i <= R; i++) {
        freqPrime[i] = freqPrime[i - 1];
 
        // If the frequency of primes
        // is a prime
        if (!prime[countPrime[i]]) {
 
            // Increase count of
            // required numbers
            freqPrime[i]++;
        }
    }
 
 
    // Return the required count
    return (freqPrime[R] -
            freqPrime[L - 1]);
}
 
    // Driver Code
     
    // Given range
    let L = 4, R = 12;
 
    // Function call
    document.write(count_crazy_primes(L, R));
 

输出：

5

时间复杂度： O(R*log(log(R)))
辅助空间： O(R)