


'''simple docstring'''

from math import isqrt, loga






def __UpperCamelCase   (			__lowerCamelCase		: int				)      ->     list[int]:







     '''simple docstring'''






     _a						=		[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						=		False

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






def __UpperCamelCase   (			__lowerCamelCase		: int = 80_0800      ,  __lowerCamelCase		: int = 80_0800				)      ->     int:







     '''simple docstring'''






     _a						=		degree * loga(__lowerCamelCase				)
     _a						=		int(__lowerCamelCase				)
     _a						=		calculate_prime_numbers(__lowerCamelCase				)

     _a						=		0
     _a						=		0
     _a						=		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() = }''')





'''simple docstring'''

from math import factorial, pi






def __UpperCamelCase   (			__lowerCamelCase		: float      ,  __lowerCamelCase		: int = 30				)      ->     float:







     '''simple docstring'''






     if not isinstance(__lowerCamelCase      ,  (int, float)				):
          raise ValueError("maclaurin_sin() requires either an int or float for theta"				)

     if not isinstance(__lowerCamelCase      ,  __lowerCamelCase				) or accuracy <= 0:
          raise ValueError("maclaurin_sin() requires a positive int for accuracy"				)

     _a						=		float(__lowerCamelCase				)
     _a						=		theta // (2 * pi)
     theta -= 2 * div * pi
     return sum(
         (-1) ** r * theta ** (2 * r + 1) / factorial(2 * r + 1				) for r in range(__lowerCamelCase				)				)






def __UpperCamelCase   (			__lowerCamelCase		: float      ,  __lowerCamelCase		: int = 30				)      ->     float:







     '''simple docstring'''






     if not isinstance(__lowerCamelCase      ,  (int, float)				):
          raise ValueError("maclaurin_cos() requires either an int or float for theta"				)

     if not isinstance(__lowerCamelCase      ,  __lowerCamelCase				) or accuracy <= 0:
          raise ValueError("maclaurin_cos() requires a positive int for accuracy"				)

     _a						=		float(__lowerCamelCase				)
     _a						=		theta // (2 * pi)
     theta -= 2 * div * pi
     return sum((-1) ** r * theta ** (2 * r) / factorial(2 * r				) for r in range(__lowerCamelCase				)				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()

   print(maclaurin_sin(10))
   print(maclaurin_sin(-10))
   print(maclaurin_sin(10, 15))
   print(maclaurin_sin(-10, 15))

   print(maclaurin_cos(5))
   print(maclaurin_cos(-5))
   print(maclaurin_cos(10, 15))
   print(maclaurin_cos(-10, 15))

