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++ 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 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 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# 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" );
}
} |
<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)