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 Pn 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> |
Yes
Time complexity: O(sqrt(n))
Auxiliary space: O(1)
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