

import math






def    __lowercase (       _A     )       ->  list:
      if n <= 0:
            raise ValueError("""Only positive integers have prime factors"""     )
      SCREAMING_SNAKE_CASE	: Optional[int]					       =				[]
      while n % 2 == 0:
            pf.append(2     )
            SCREAMING_SNAKE_CASE	: Optional[int]					       =				int(n / 2     )
      for i in range(3  ,      int(math.sqrt(_A     )     ) + 1  ,      2     ):
            while n % i == 0:
                  pf.append(_A     )
                  SCREAMING_SNAKE_CASE	: Dict					       =				int(n / i     )
      if n > 2:
            pf.append(_A     )
      return pf






def    __lowercase (       _A     )       ->  int:
      if n <= 0:
            raise ValueError("""Only positive numbers are accepted"""     )
      SCREAMING_SNAKE_CASE	: List[str]					       =				1
      SCREAMING_SNAKE_CASE	: List[Any]					       =				1
      while n % 2 == 0:
            temp += 1
            SCREAMING_SNAKE_CASE	: str					       =				int(n / 2     )
      div *= temp
      for i in range(3  ,      int(math.sqrt(_A     )     ) + 1  ,      2     ):
            SCREAMING_SNAKE_CASE	: Optional[Any]					       =				1
            while n % i == 0:
                  temp += 1
                  SCREAMING_SNAKE_CASE	: Any					       =				int(n / i     )
            div *= temp
      if n > 1:
            div *= 2
      return div






def    __lowercase (       _A     )       ->  int:
      if n <= 0:
            raise ValueError("""Only positive numbers are accepted"""     )
      SCREAMING_SNAKE_CASE	: int					       =				1
      SCREAMING_SNAKE_CASE	: List[Any]					       =				1
      while n % 2 == 0:
            temp += 1
            SCREAMING_SNAKE_CASE	: Dict					       =				int(n / 2     )
      if temp > 1:
            s *= (2**temp - 1) / (2 - 1)
      for i in range(3  ,      int(math.sqrt(_A     )     ) + 1  ,      2     ):
            SCREAMING_SNAKE_CASE	: int					       =				1
            while n % i == 0:
                  temp += 1
                  SCREAMING_SNAKE_CASE	: Dict					       =				int(n / i     )
            if temp > 1:
                  s *= (i**temp - 1) / (i - 1)
      return int(_A     )






def    __lowercase (       _A     )       ->  int:
      SCREAMING_SNAKE_CASE	: List[Any]					       =				n
      for x in set(prime_factors(_A     )     ):
            s *= (x - 1) / x
      return int(_A     )


if __name__ == "__main__":
  import doctest

  doctest.testmod()







import inspect
import os
import unittest

import torch

import accelerate
from accelerate import debug_launcher
from accelerate.test_utils import (
    execute_subprocess_async,
    require_cpu,
    require_multi_gpu,
    require_single_gpu,
    test_sync,
)
from accelerate.utils import patch_environment







class     a__    (				unittest.TestCase    ):







  """simple docstring"""




  def 	_lowercase (     self      :	Optional[int]						)   ->Tuple:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: Optional[Any]					       =				inspect.getfile(accelerate.test_utils						)
        SCREAMING_SNAKE_CASE	: int					       =				os.path.sep.join(mod_file.split(os.path.sep						)[:-1] + ["""scripts""", """test_sync.py"""]						)


  @require_cpu
  def 	_lowercase (     self      :	Optional[int]						)   ->str:
        """simple docstring"""







        debug_launcher(test_sync.main			,			num_processes=1						)


  @require_cpu
  def 	_lowercase (     self      :	Optional[int]						)   ->int:
        """simple docstring"""







        debug_launcher(test_sync.main						)


  @require_single_gpu
  def 	_lowercase (     self      :	str						)   ->Any:
        """simple docstring"""







        test_sync.main()




  @require_multi_gpu
  def 	_lowercase (     self      :	Union[str, Any]						)   ->List[Any]:
        """simple docstring"""







        print(f"Found {torch.cuda.device_count()} devices."						)
        SCREAMING_SNAKE_CASE	: List[str]					       =				["""torchrun""", f"--nproc_per_node={torch.cuda.device_count()}", self.test_file_path]
        with patch_environment(omp_num_threads=1						):
              execute_subprocess_async(UpperCAmelCase__			,			env=os.environ.copy()						)




