





from math import isqrt, loga



def      __lowercase     (   _SCREAMING_SNAKE_CASE			)				->     list[int]:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE	=		[True] * max_number
      for i in range(2				,		isqrt(max_number - 1			) + 1			):
            if is_prime[i]:
                  for j in range(i**2				,		_SCREAMING_SNAKE_CASE				,		_SCREAMING_SNAKE_CASE			):
                        SCREAMING_SNAKE_CASE	=		False

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



def      __lowercase     (   _SCREAMING_SNAKE_CASE = 80_08_00				,		_SCREAMING_SNAKE_CASE = 80_08_00			)				->     int:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE	=		degree * loga(_SCREAMING_SNAKE_CASE			)
      SCREAMING_SNAKE_CASE	=		int(_SCREAMING_SNAKE_CASE			)
      SCREAMING_SNAKE_CASE	=		calculate_prime_numbers(_SCREAMING_SNAKE_CASE			)

      SCREAMING_SNAKE_CASE	=		0
      SCREAMING_SNAKE_CASE	=		0
      SCREAMING_SNAKE_CASE	=		len(_SCREAMING_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() = }''')









# We ignore warnings about stepping the scheduler since we step it ourselves during gradient accumulation

import warnings

from .state import AcceleratorState, GradientState


warnings.filterwarnings("""ignore""", category=UserWarning, module="""torch.optim.lr_scheduler""")

class 	UpperCamelCase__				:



     '''simple docstring'''

     def __init__(							self		:					Optional[Any]		,lowerCamelCase__		:					Optional[int]		,lowerCamelCase__		:					Tuple		,lowerCamelCase__		:					bool = True		,lowerCamelCase__		:					bool = False )  ->	Tuple:




           '''simple docstring'''


           SCREAMING_SNAKE_CASE	=		scheduler
           SCREAMING_SNAKE_CASE	=		optimizers if isinstance(lowerCamelCase__		,(list, tuple) ) else [optimizers]
           SCREAMING_SNAKE_CASE	=		split_batches
           SCREAMING_SNAKE_CASE	=		step_with_optimizer
           SCREAMING_SNAKE_CASE	=		GradientState()






     def        SCREAMING_SNAKE_CASE__						(							self		:					str		,*lowerCamelCase__		:					Union[str, Any]		,**lowerCamelCase__		:					Optional[int] )  ->	Dict:




           '''simple docstring'''


           if not self.step_with_optimizer:
                 # No link between scheduler and optimizer -> just step
                 self.scheduler.step(*lowerCamelCase__		,**lowerCamelCase__ )
                 return

           # Otherwise, first make sure the optimizer was stepped.
           if not self.gradient_state.sync_gradients:
                 if self.gradient_state.adjust_scheduler:
                       self.scheduler._step_count += 1
                 return

           for opt in self.optimizers:
                 if opt.step_was_skipped:
                       return
           if self.split_batches:
                 # Split batches -> the training dataloader batch size is not changed so one step per training step
                 self.scheduler.step(*lowerCamelCase__		,**lowerCamelCase__ )
           else:
                 # Otherwise the training dataloader batch size was multiplied by `num_processes`, so we need to do
                 # num_processes steps per training step
                 SCREAMING_SNAKE_CASE	=		AcceleratorState().num_processes
                 for _ in range(lowerCamelCase__ ):
                       # Special case when using OneCycle and `drop_last` was not used
                       if hasattr(self.scheduler		,"""total_steps""" ):
                             if self.scheduler._step_count <= self.scheduler.total_steps:
                                   self.scheduler.step(*lowerCamelCase__		,**lowerCamelCase__ )
                       else:
                             self.scheduler.step(*lowerCamelCase__		,**lowerCamelCase__ )






     def        SCREAMING_SNAKE_CASE__						(							self		:					int )  ->	Optional[Any]:




           '''simple docstring'''


           return self.scheduler.get_last_lr()






     def        SCREAMING_SNAKE_CASE__						(							self		:					List[Any] )  ->	Optional[Any]:




           '''simple docstring'''


           return self.scheduler.state_dict()






     def        SCREAMING_SNAKE_CASE__						(							self		:					Optional[int]		,lowerCamelCase__		:					str )  ->	int:




           '''simple docstring'''


           self.scheduler.load_state_dict(lowerCamelCase__ )






     def        SCREAMING_SNAKE_CASE__						(							self		:					List[Any] )  ->	Optional[Any]:




           '''simple docstring'''


           return self.scheduler.get_lr()






     def        SCREAMING_SNAKE_CASE__						(							self		:					Dict		,*lowerCamelCase__		:					int		,**lowerCamelCase__		:					Tuple )  ->	Optional[int]:




           '''simple docstring'''


           return self.scheduler.print_lr(*lowerCamelCase__		,**lowerCamelCase__ )


