Check if N is a Weak Prime number or not

Given a positive integer N, the task is to check if N is a weak Prime or not.

In number theory, a weak prime is a prime number that is less than the arithmetic mean of nearest prime numbers i.e next and previous prime numbers.
First few weak prime numbers are 3, 7, 13, 19, 23, 31, 43, 47, 61, …
A weak prime P_n can be represented as-

where n is its index in the ordered set of prime numbers.

Examples:

Input: N = 13
Output: Yes
13 is 6th prime number, the arithmetic mean of 5th and 7th prime number i.e. 11 and 17 is 14.
13 is less than 14 so 13 is a weak prime.

Input: N = 11
Output: No

Approach:

• If N is not a prime number or it is the first prime number i.e. 2 then print No.
• Else find the primes closest to N (one on the left and one on the right) and store their arithmetic mean in mean.
  • If N < mean then print Yes.
  • Else print No.

Below is the implementation of the above approach:

C++14

// C++ program to check
// if a given number is weak prime
#include <bits/stdc++.h>
using namespace std;

// Utility function to check
// if a number is prime or not
bool isPrime(int n)
{

    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;

    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;

    for(int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Function that returns true
// if n is a weak prime
bool isWeakPrime(int n)
{

    // If n is not a prime number or
    // n is the first prime then return false
    if (!isPrime(n) || n == 2)
        return false;

    // Initialize previous_prime to n - 1
    // and next_prime to n + 1
    int previous_prime = n - 1;
    int next_prime = n + 1;

    // Find next prime number
    while (!isPrime(next_prime))
        next_prime++;

    // Find previous prime number
    while (!isPrime(previous_prime))
        previous_prime--;

    // Arithmetic mean
    int mean = (previous_prime +
                next_prime) / 2;

    // If n is a weak prime
    if (n < mean)
        return true;
    else
        return false;
}

// Driver code
int main()
{
    int n = 13;

    if (isWeakPrime(n))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

// This code is contributed by himanshu77

Java

// Java program to check
// if a given number is weak prime
import java.util.*;
class GFG{

// Utility function to check
// if a number is prime or not
static boolean isPrime(int n)
{
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;

    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;

    for (int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Function that returns true
// if n is a weak prime
static boolean isWeakPrime(int n)
{
    // If n is not a prime number or
    // n is the first prime then return false
    if (!isPrime(n) || n == 2)
        return false;

    // Initialize previous_prime to n - 1
    // and next_prime to n + 1
    int previous_prime = n - 1;
    int next_prime = n + 1;

    // Find next prime number
    while (!isPrime(next_prime))
        next_prime++;

    // Find previous prime number
    while (!isPrime(previous_prime))
        previous_prime--;

    // Arithmetic mean
    int mean = (previous_prime +
                next_prime) / 2;

    // If n is a weak prime
    if (n < mean)
        return true;
    else
        return false;
}

// Driver code
public static void main(String args[])
{
    int n = 13;

    if (isWeakPrime(n))
        System.out.print("Yes");
    else
        System.out.print("No");
}
}

// This code is contributed by Code_Mech

Python3

# Python3 program to check if a given
# number is weak prime

# Utility function to check
# if a number is prime or not
def isPrime(n):

    # Corner cases
    if (n <= 1):
        return False
    if (n <= 3):
        return True

    # This is checked so that we can skip
    # middle five numbers in below loop
    if (n % 2 == 0 or n % 3 == 0):
        return False

    i = 5
    while (i * i <= n):
        if (n % i == 0 or n % (i + 2) == 0):
            return False
        i = i + 6

    return True

# Function that returns true
# if n is a weak prime
def isWeakPrime(n):

    # declaring variables as global
    global next_prime, previous_prime

    # If n is not a prime number or n is
    # the first prime then return false
    if(not isPrime(n) or n == 2):
        return False

    # Initialize previous_prime to n - 1
    # and next_prime to n + 1
    previous_prime = n - 1
    next_prime = n + 1

    # Find next prime number
    while(not isPrime(next_prime)):
        next_prime += 1

    # Find previous prime number
    while (not isPrime(previous_prime)):
        previous_prime -= 1

    # Arithmetic mean
    mean = (previous_prime + next_prime) // 2

    # If n is a weak prime
    if(n < mean):
        return True
    else:
        return False

# Driver code
if __name__ == '__main__':

    n = 13

    if(isWeakPrime(n)):
        print("Yes")
    else:
        print("No")

# This code is contributed by Shivam Singh

C#

// C# program to check if a given number is weak prime
using System;
class GFG {

    // Utility function to check
    // if a number is prime or not
    static bool isPrime(int n)
    {
        // Corner cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;

        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
            return false;

        for (int i = 5; i * i <= n; i = i + 6)
            if (n % i == 0 || n % (i + 2) == 0)
                return false;

        return true;
    }

    // Function that returns true
    // if n is a weak prime
    static bool isWeakPrime(int n)
    {
        // If n is not a prime number or
        // n is the first prime then return false
        if (!isPrime(n) || n == 2)
            return false;

        // Initialize previous_prime to n - 1
        // and next_prime to n + 1
        int previous_prime = n - 1;
        int next_prime = n + 1;

        // Find next prime number
        while (!isPrime(next_prime))
            next_prime++;

        // Find previous prime number
        while (!isPrime(previous_prime))
            previous_prime--;

        // Arithmetic mean
        int mean = (previous_prime
                    + next_prime)
                   / 2;

        // If n is a weak prime
        if (n < mean)
            return true;
        else
            return false;
    }

    // Driver code
    public static void Main()
    {
        int n = 13;

        if (isWeakPrime(n))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}

Javascript

<script>
    // Javascript program to check
    // if a given number is weak prime

    // Utility function to check
    // if a number is prime or not
    function isPrime(n)
    {

        // Corner cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;

        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
            return false;

        for(let i = 5; i * i <= n; i = i + 6)
            if (n % i == 0 || n % (i + 2) == 0)
                return false;

        return true;
    }

    // Function that returns true
    // if n is a weak prime
    function isWeakPrime(n)
    {

        // If n is not a prime number or
        // n is the first prime then return false
        if (!isPrime(n) || n == 2)
            return false;

        // Initialize previous_prime to n - 1
        // and next_prime to n + 1
        let previous_prime = n - 1;
        let next_prime = n + 1;

        // Find next prime number
        while (!isPrime(next_prime))
            next_prime++;

        // Find previous prime number
        while (!isPrime(previous_prime))
            previous_prime--;

        // Arithmetic mean
        let mean = (previous_prime +
                    next_prime) / 2;

        // If n is a weak prime
        if (n < mean)
            return true;
        else
            return false;
    }

    let n = 13;

    if (isWeakPrime(n))
        document.write("Yes");
    else
        document.write("No");

// This code is contributed by divyesh072019.
</script>

Output

Yes

Time complexity: O(sqrt(n))
Auxiliary space: O(1)