


from functools import lru_cache






def 							__lowercase     (				__lowerCAmelCase							:       int     ):
							a__    =  2
							a__    =  set()
							while i * i <= n:
														if n % i:
																					i += 1
														else:
																					n //= i
																					factors.add(__lowerCAmelCase     )
							if n > 1:
														factors.add(__lowerCAmelCase     )
							return factors






@lru_cache
def 							__lowercase     (				__lowerCAmelCase							:       int     ):
							return len(unique_prime_factors(__lowerCAmelCase     )     )






def 							__lowercase     (				__lowerCAmelCase							:       list     ):
							return len(set(__lowerCAmelCase     )     ) in (0, 1)






def 							__lowercase     (				__lowerCAmelCase							:       int     ):
							a__    =  2

							while True:
														# Increment each value of a generated range
														a__    =  [base + i for i in range(__lowerCAmelCase     )]

														# Run elements through out unique_prime_factors function
														# Append our target number to the end.
														a__    =  [upf_len(__lowerCAmelCase     ) for x in group]
														checker.append(__lowerCAmelCase     )

														# If all numbers in the list are equal, return the group variable.
														if equality(__lowerCAmelCase     ):
																					return group

														# Increment our base variable by 1
														base += 1






def 							__lowercase     (				__lowerCAmelCase							:       int = 4     ):
							a__    =  run(__lowerCAmelCase     )
							return results[0] if len(__lowerCAmelCase     ) else None


if __name__ == "__main__":
						print(solution())








import math
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils import SchedulerMixin, SchedulerOutput







class  snake_case_     (lowerCamelCase_			,		lowerCamelCase_   ):
					UpperCAmelCase__      :					Optional[Any]					   =    1
					@register_to_config
					def __init__(   self     :Optional[int]    ,__snake_case     :int = 10_00    ,__snake_case     :Optional[Union[np.ndarray, List[float]]] = None   )      ->					int:
												# set `betas`, `alphas`, `timesteps`
												self.set_timesteps(__snake_case   )

												# standard deviation of the initial noise distribution
												a__    =  1.0

												# For now we only support F-PNDM, i.e. the runge-kutta method
												# For more information on the algorithm please take a look at the paper: https://arxiv.org/pdf/2202.09778.pdf
												# mainly at formula (9), (12), (13) and the Algorithm 2.
												a__    =  4

												# running values
												a__    =  []
					def 				lowerCamelCase__(   self     :Union[str, Any]    ,__snake_case     :int    ,__snake_case     :Union[str, torch.device] = None   )      ->					Union[str, Any]:
												a__    =  num_inference_steps
												a__    =  torch.linspace(1    ,0    ,num_inference_steps + 1   )[:-1]
												a__    =  torch.cat([steps, torch.tensor([0.0]   )]   )

												if self.config.trained_betas is not None:
																			a__    =  torch.tensor(self.config.trained_betas    ,dtype=torch.floataa   )
												else:
																			a__    =  torch.sin(steps * math.pi / 2   ) ** 2

												a__    =  (1.0 - self.betas**2) ** 0.5

												a__    =  (torch.atana(self.betas    ,self.alphas   ) / math.pi * 2)[:-1]
												a__    =  timesteps.to(__snake_case   )

												a__    =  []
					def 				lowerCamelCase__(   self     :Any    ,__snake_case     :torch.FloatTensor    ,__snake_case     :int    ,__snake_case     :torch.FloatTensor    ,__snake_case     :bool = True    ,)      ->					Union[SchedulerOutput, Tuple]:
												if self.num_inference_steps is None:
																			raise ValueError(
																			    'Number of inference steps is \'None\', you need to run \'set_timesteps\' after creating the scheduler'   )

												a__    =  (self.timesteps == timestep).nonzero().item()
												a__    =  timestep_index + 1

												a__    =  sample * self.betas[timestep_index] + model_output * self.alphas[timestep_index]
												self.ets.append(__snake_case   )

												if len(self.ets   ) == 1:
																			a__    =  self.ets[-1]
												elif len(self.ets   ) == 2:
																			a__    =  (3 * self.ets[-1] - self.ets[-2]) / 2
												elif len(self.ets   ) == 3:
																			a__    =  (23 * self.ets[-1] - 16 * self.ets[-2] + 5 * self.ets[-3]) / 12
												else:
																			a__    =  (1 / 24) * (55 * self.ets[-1] - 59 * self.ets[-2] + 37 * self.ets[-3] - 9 * self.ets[-4])

												a__    =  self._get_prev_sample(__snake_case    ,__snake_case    ,__snake_case    ,__snake_case   )

												if not return_dict:
																			return (prev_sample,)

												return SchedulerOutput(prev_sample=__snake_case   )
					def 				lowerCamelCase__(   self     :Union[str, Any]    ,__snake_case     :torch.FloatTensor    ,*__snake_case     :int    ,**__snake_case     :Optional[int]   )      ->					torch.FloatTensor:
												return sample
					def 				lowerCamelCase__(   self     :Optional[Any]    ,__snake_case     :List[Any]    ,__snake_case     :Optional[int]    ,__snake_case     :Dict    ,__snake_case     :Any   )      ->					Optional[Any]:
												a__    =  self.alphas[timestep_index]
												a__    =  self.betas[timestep_index]

												a__    =  self.alphas[prev_timestep_index]
												a__    =  self.betas[prev_timestep_index]

												a__    =  (sample - sigma * ets) / max(__snake_case    ,1E-8   )
												a__    =  next_alpha * pred + ets * next_sigma

												return prev_sample



					def __len__(   self     :Any   )      ->					Union[str, Any]:
												return self.config.num_train_timesteps




