
import math



def 			UpperCamelCase       (				lowerCAmelCase__						):
					'''simple docstring'''
					if n <= 0:
										raise ValueError('''Only positive integers have prime factors'''						)
					lowercase		      =		[]
					while n % 2 == 0:
										pf.append(2						)
										lowercase		      =		int(n / 2						)
					for i in range(3					,     int(math.sqrt(lowerCAmelCase__						)						) + 1					,     2						):
										while n % i == 0:
															pf.append(lowerCAmelCase__						)
															lowercase		      =		int(n / i						)
					if n > 2:
										pf.append(lowerCAmelCase__						)
					return pf



def 			UpperCamelCase       (				lowerCAmelCase__						):
					'''simple docstring'''
					if n <= 0:
										raise ValueError('''Only positive numbers are accepted'''						)
					lowercase		      =		1
					lowercase		      =		1
					while n % 2 == 0:
										temp += 1
										lowercase		      =		int(n / 2						)
					div *= temp
					for i in range(3					,     int(math.sqrt(lowerCAmelCase__						)						) + 1					,     2						):
										lowercase		      =		1
										while n % i == 0:
															temp += 1
															lowercase		      =		int(n / i						)
										div *= temp
					if n > 1:
										div *= 2
					return div



def 			UpperCamelCase       (				lowerCAmelCase__						):
					'''simple docstring'''
					if n <= 0:
										raise ValueError('''Only positive numbers are accepted'''						)
					lowercase		      =		1
					lowercase		      =		1
					while n % 2 == 0:
										temp += 1
										lowercase		      =		int(n / 2						)
					if temp > 1:
										s *= (2**temp - 1) / (2 - 1)
					for i in range(3					,     int(math.sqrt(lowerCAmelCase__						)						) + 1					,     2						):
										lowercase		      =		1
										while n % i == 0:
															temp += 1
															lowercase		      =		int(n / i						)
										if temp > 1:
															s *= (i**temp - 1) / (i - 1)
					return int(lowerCAmelCase__						)



def 			UpperCamelCase       (				lowerCAmelCase__						):
					'''simple docstring'''
					lowercase		      =		n
					for x in set(prime_factors(lowerCAmelCase__						)						):
										s *= (x - 1) / x
					return int(lowerCAmelCase__						)


if __name__ == "__main__":
							import doctest

							doctest.testmod()






import torch

from accelerate import Accelerator, DistributedType







class 							lowercase						:




							def __enter__(							self):
												if self.enabled:
																	lowercase		      =		self.model.no_sync()
																	self.model_sync_obj.__enter__()

												return self




							def __exit__(							self  ,A__  ,A__  ,A__):
												if self.enabled:
																	# Average all models on exit
																	self._sync_and_avg_model_params()
																	self.model_sync_obj.__exit__(A__  ,A__  ,A__)




							def __init__(							self  ,A__  ,A__  ,A__  ,A__ = True):
												if accelerator.distributed_type not in [
												    DistributedType.NO,
												    DistributedType.MULTI_CPU,
												    DistributedType.MULTI_GPU,
												]:
																	raise NotImplementedError('''LocalSGD is supported only for CPUs and GPUs (no DeepSpeed or MegatronLM)''')
												lowercase		      =		enabled and accelerator.distributed_type != DistributedType.NO
												lowercase		      =		0
												if self.enabled:
																	lowercase		      =		accelerator
																	lowercase		      =		model
																	lowercase		      =		local_sgd_steps




							def 					A__ (							self):
												self.num_steps += 1
												if not self.enabled:
																	return

												if self.num_steps % self.local_sgd_steps == 0:
																	self._sync_and_avg_model_params()





							def 					A__ (							self):
												self.accelerator.wait_for_everyone()
												with self.accelerator.autocast():
																	for param in self.model.parameters():
																						lowercase		      =		self.accelerator.reduce(param.data  ,reduction='''mean''')




