



"""simple docstring"""
import math
import unittest

def       _snake_case						(						UpperCamelCase						:		int						):
 assert isinstance(snake_case__			,		snake_case__						) and (
     number >= 0
 ), "'number' must been an int and positive"

 if 1 < number < 4:
  # 2 and 3 are primes
  return True
 elif number < 2 or number % 2 == 0 or number % 3 == 0:
  # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False

 # All primes number are in format of 6k +/- 1
 for i in range(5			,		int(math.sqrt(snake_case__						) + 1						)			,		6						):
  if number % i == 0 or number % (i + 2) == 0:
   return False
 return True



class       SCREAMING_SNAKE_CASE__  (					unittest.TestCase ):

   def 	SCREAMING_SNAKE_CASE     (   self						)  ->      Union[str, Any]:





    '''simple docstring'''




    self.assertTrue(is_prime(2						)						)
    self.assertTrue(is_prime(3						)						)
    self.assertTrue(is_prime(5						)						)
    self.assertTrue(is_prime(7						)						)
    self.assertTrue(is_prime(11						)						)
    self.assertTrue(is_prime(13						)						)
    self.assertTrue(is_prime(17						)						)
    self.assertTrue(is_prime(19						)						)
    self.assertTrue(is_prime(23						)						)
    self.assertTrue(is_prime(29						)						)




   def 	SCREAMING_SNAKE_CASE     (   self						)  ->      int:





    '''simple docstring'''




    with self.assertRaises(lowerCAmelCase_						):
     is_prime(-19						)
    self.assertFalse(
        is_prime(0						)      ,     """Zero doesn't have any positive factors, primes must have exactly two."""      ,     )
    self.assertFalse(
        is_prime(1						)      ,     """One only has 1 positive factor, primes must have exactly two."""      ,     )
    self.assertFalse(is_prime(2 * 2						)						)
    self.assertFalse(is_prime(2 * 3						)						)
    self.assertFalse(is_prime(3 * 3						)						)
    self.assertFalse(is_prime(3 * 5						)						)
    self.assertFalse(is_prime(3 * 5 * 7						)						)


if __name__ == "__main__":
      unittest.main()

from math import isqrt, loga






def  snake_case  (       snake_case__			:int)						->   list[int]:
       _A        =  [True] * max_number
       for i in range(2   ,	isqrt(max_number - 1) + 1):
              if is_prime[i]:
                     for j in range(i**2   ,	snake_case__   ,	snake_case__):
                            _A        =  False

       return [i for i in range(2   ,	snake_case__) if is_prime[i]]






def  snake_case  (       snake_case__			:int = 800_800   ,	snake_case__			:int = 800_800)						->   int:
       _A        =  degree * loga(snake_case__)
       _A        =  int(snake_case__)
       _A        =  calculate_prime_numbers(snake_case__)

       _A        =  0
       _A        =  0
       _A        =  len(snake_case__) - 1
       while left < right:
              while (
                  prime_numbers[right] * loga(prime_numbers[left])
                  + prime_numbers[left] * loga(prime_numbers[right])
                  > upper_bound
              ):
                     right -= 1
              hybrid_integers_count += right - left
              left += 1

       return hybrid_integers_count


if __name__ == "__main__":
    print(F'''{solution() = }''')




