





from math import isqrt, loga



def      _lowerCAmelCase		(	_lowerCAmelCase  ):
   '''simple docstring'''


   A_  :					Dict         =						[True] * max_number
   for i in range(2	,isqrt(max_number - 1  ) + 1  ):
      if is_prime[i]:
         for j in range(i**2	,_lowerCAmelCase	,_lowerCAmelCase  ):
            A_  :					List[str]         =						False

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



def      _lowerCAmelCase		(	_lowerCAmelCase = 8_0_0_8_0_0	,_lowerCAmelCase = 8_0_0_8_0_0  ):
   '''simple docstring'''


   A_  :					Optional[int]         =						degree * loga(_lowerCAmelCase  )
   A_  :					Any         =						int(_lowerCAmelCase  )
   A_  :					Optional[int]         =						calculate_prime_numbers(_lowerCAmelCase  )

   A_  :					Tuple         =						0
   A_  :					List[Any]         =						0
   A_  :					int         =						len(_lowerCAmelCase  ) - 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() = }''')














from typing import Any

import numpy as np
def 		_a       ( a					:np.ndarray					)				->     bool:
 return np.array_equal(a      ,				matrix.conjugate().T					)
def 		_a       ( a					:np.ndarray      ,				a					:np.ndarray					)				->     Any:
 a						    = v.conjugate().T
 a						    = v_star.dot(a					)
 assert isinstance(a      ,				np.ndarray					)
 return (v_star_dot.dot(a					)) / (v_star.dot(a					))
def 		_a       ( )				->     None:
 a						    = np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]					)
 a						    = np.array([[1], [2], [3]]					)
 assert is_hermitian(a					), F"""{a} is not hermitian."""
 print(rayleigh_quotient(a      ,				a					)					)

 a						    = np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]					)
 assert is_hermitian(a					), F"""{a} is not hermitian."""
 assert rayleigh_quotient(a      ,				a					) == float(3					)


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     tests()


