
def 							SCREAMING_SNAKE_CASE_	(					UpperCAmelCase_	:     int = 1_0_0_0_0_0_0				)			->						int:
       SCREAMING_SNAKE_CASE_							:				str		=[i - 1 for i in range(limit + 1				)]

       for i in range(2			,							limit + 1				):
              if phi[i] == i - 1:
                     for j in range(2 * i			,							limit + 1			,							UpperCAmelCase_				):
                            phi[j] -= phi[j] // i

       return sum(phi[2 : limit + 1]				)


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


from typing import List, Optional, Tuple, Union

import torch

from ...models import UNetaDModel
from ...schedulers import KarrasVeScheduler
from ...utils import randn_tensor
from ..pipeline_utils import DiffusionPipeline, ImagePipelineOutput


class 			lowercase_		(      A       ):
       __lowerCamelCase				       =						42
       __lowerCamelCase				       =						42


       def __init__(    self						,       __A						,       __A				)						->    List[str]:
              super().__init__()
              self.register_modules(unet=__A						,       scheduler=__A				)





       @torch.no_grad()
       def __call__(    self						,       __A = 1						,       __A = 50						,       __A = None						,       __A = "pil"						,       __A = True						,       **__A						,       )						->    Union[Tuple, ImagePipelineOutput]:
              SCREAMING_SNAKE_CASE_							:				Dict		=self.unet.config.sample_size
              SCREAMING_SNAKE_CASE_							:				Tuple		=(batch_size, 3, img_size, img_size)

              SCREAMING_SNAKE_CASE_							:				int		=self.unet

              # sample x_0 ~ N(0, sigma_0^2 * I)
              SCREAMING_SNAKE_CASE_							:				Tuple		=randn_tensor(__A						,       generator=__A						,       device=self.device				) * self.scheduler.init_noise_sigma

              self.scheduler.set_timesteps(__A				)

              for t in self.progress_bar(self.scheduler.timesteps				):
                     # here sigma_t == t_i from the paper
                     SCREAMING_SNAKE_CASE_							:				Union[str, Any]		=self.scheduler.schedule[t]
                     SCREAMING_SNAKE_CASE_							:				List[Any]		=self.scheduler.schedule[t - 1] if t > 0 else 0

                     # 1. Select temporarily increased noise level sigma_hat
                     # 2. Add new noise to move from sample_i to sample_hat
                     SCREAMING_SNAKE_CASE_	,     SCREAMING_SNAKE_CASE_							:				Dict		=self.scheduler.add_noise_to_input(__A						,       __A						,       generator=__A				)

                     # 3. Predict the noise residual given the noise magnitude `sigma_hat`
                     # The model inputs and output are adjusted by following eq. (213) in [1].
                     SCREAMING_SNAKE_CASE_							:				Tuple		=(sigma_hat / 2) * model((sample_hat + 1) / 2						,       sigma_hat / 2				).sample

                     # 4. Evaluate dx/dt at sigma_hat
                     # 5. Take Euler step from sigma to sigma_prev
                     SCREAMING_SNAKE_CASE_							:				List[Any]		=self.scheduler.step(__A						,       __A						,       __A						,       __A				)

                     if sigma_prev != 0:
                            # 6. Apply 2nd order correction
                            # The model inputs and output are adjusted by following eq. (213) in [1].
                            SCREAMING_SNAKE_CASE_							:				int		=(sigma_prev / 2) * model((step_output.prev_sample + 1) / 2						,       sigma_prev / 2				).sample
                            SCREAMING_SNAKE_CASE_							:				Dict		=self.scheduler.step_correct(
                                __A						,       __A						,       __A						,       __A						,       step_output.prev_sample						,       step_output['''derivative''']						,       )
                     SCREAMING_SNAKE_CASE_							:				Any		=step_output.prev_sample

              SCREAMING_SNAKE_CASE_							:				List[Any]		=(sample / 2 + 0.5).clamp(0						,       1				)
              SCREAMING_SNAKE_CASE_							:				Any		=sample.cpu().permute(0						,       2						,       3						,       1				).numpy()
              if output_type == "pil":
                     SCREAMING_SNAKE_CASE_							:				Optional[Any]		=self.numpy_to_pil(__A				)

              if not return_dict:
                     return (image,)

              return ImagePipelineOutput(images=__A				)
