



UpperCamelCase_									=   {
    '''0''': 1,
    '''1''': 1,
    '''2''': 2,
    '''3''': 6,
    '''4''': 24,
    '''5''': 120,
    '''6''': 720,
    '''7''': 5040,
    '''8''': 40320,
    '''9''': 362880,
}

UpperCamelCase_									=   {145: 145}

UpperCamelCase_									=   {
    145: 0,
    169: 3,
    36301: 3,
    1454: 3,
    871: 2,
    45361: 2,
    872: 2,
}







def       lowerCamelCase_  (    _a       :						int    ):



    '''simple docstring'''




    if n in CACHE_SUM_DIGIT_FACTORIALS:
        return CACHE_SUM_DIGIT_FACTORIALS[n]
    UpperCAmelCase_     :       Union[str, Any]  		=					sum(DIGIT_FACTORIALS[let] for let in str(_a    )    )
    UpperCAmelCase_     :       Any  		=					ret
    return ret







def       lowerCamelCase_  (    _a       :						int   ,		_a       :						set | None = None    ):



    '''simple docstring'''




    UpperCAmelCase_     :       Dict  		=					previous or set()
    if n in CHAIN_LENGTH_CACHE:
        return CHAIN_LENGTH_CACHE[n]
    UpperCAmelCase_     :       List[Any]  		=					sum_digit_factorials(_a    )
    if next_number in previous:
        UpperCAmelCase_     :       int  		=					0
        return 0
    else:
        previous.add(_a    )
        UpperCAmelCase_     :       Optional[Any]  		=					1 + chain_length(_a   ,		_a    )
        UpperCAmelCase_     :       Any  		=					ret
        return ret







def       lowerCamelCase_  (    _a       :						int = 60   ,		_a       :						int = 100_0000    ):



    '''simple docstring'''




    return sum(1 for i in range(1   ,		_a    ) if chain_length(_a    ) == num_terms    )


if __name__ == "__main__":
      print(F"{solution() = }")







"""simple docstring"""


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

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import randn_tensor
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput





def 				a__      (	_UpperCAmelCase						,  _UpperCAmelCase=0.999						,  _UpperCAmelCase="cosine"						,  )     ->	str:

 '''simple docstring'''






 if alpha_transform_type == "cosine":

  def alpha_bar_fn(_UpperCAmelCase						):
   return math.cos((t + 0.008) / 1.008 * math.pi / 2						) ** 2

 elif alpha_transform_type == "exp":

  def alpha_bar_fn(_UpperCAmelCase						):
   return math.exp(t * -12.0						)

 else:
  raise ValueError(f"Unsupported alpha_tranform_type: {alpha_transform_type}"						)

 snake_case__     				=					[]
 for i in range(_UpperCAmelCase						):
  snake_case__     				=					i / num_diffusion_timesteps
  snake_case__     				=					(i + 1) / num_diffusion_timesteps
  betas.append(min(1 - alpha_bar_fn(_UpperCAmelCase						) / alpha_bar_fn(_UpperCAmelCase						)						,  _UpperCAmelCase						)						)
 return torch.tensor(_UpperCAmelCase						,  dtype=torch.floataa						)



class   lowercase_					(  __lowerCAmelCase					,__lowerCAmelCase				):
       snake_case__					:				Tuple									=  [e.name for e in KarrasDiffusionSchedulers]
       snake_case__					:				Any									=  2

       @register_to_config
       def __init__(      self ,       __SCREAMING_SNAKE_CASE = 1000 ,       __SCREAMING_SNAKE_CASE = 0.00085 ,       __SCREAMING_SNAKE_CASE = 0.012 ,       __SCREAMING_SNAKE_CASE = "linear" ,       __SCREAMING_SNAKE_CASE = None ,       __SCREAMING_SNAKE_CASE = "epsilon" ,       __SCREAMING_SNAKE_CASE = "linspace" ,       __SCREAMING_SNAKE_CASE = 0 ,       ):






        '''simple docstring'''

        if trained_betas is not None:
         snake_case__     				=					torch.tensor(__SCREAMING_SNAKE_CASE ,       dtype=torch.floataa  )
        elif beta_schedule == "linear":
         snake_case__     				=					torch.linspace(__SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       dtype=torch.floataa  )
        elif beta_schedule == "scaled_linear":
         # this schedule is very specific to the latent diffusion model.
         snake_case__     				=					(
             torch.linspace(beta_start**0.5 ,       beta_end**0.5 ,       __SCREAMING_SNAKE_CASE ,       dtype=torch.floataa  ) ** 2
         )
        elif beta_schedule == "squaredcos_cap_v2":
         # Glide cosine schedule
         snake_case__     				=					betas_for_alpha_bar(__SCREAMING_SNAKE_CASE  )
        else:
         raise NotImplementedError(F"{beta_schedule} does is not implemented for {self.__class__}"  )

        snake_case__     				=					1.0 - self.betas
        snake_case__     				=					torch.cumprod(self.alphas ,       dim=0  )

        #  set all values
        self.set_timesteps(__SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE  )

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE=None  ):






        '''simple docstring'''

        if schedule_timesteps is None:
         snake_case__     				=					self.timesteps

        snake_case__     				=					(schedule_timesteps == timestep).nonzero()

        # The sigma index that is taken for the **very** first `step`
        # is always the second index (or the last index if there is only 1)
        # This way we can ensure we don't accidentally skip a sigma in
        # case we start in the middle of the denoising schedule (e.g. for image-to-image)
        if len(self._index_counter  ) == 0:
         snake_case__     				=					1 if len(__SCREAMING_SNAKE_CASE  ) > 1 else 0
        else:
         snake_case__     				=					timestep.cpu().item() if torch.is_tensor(__SCREAMING_SNAKE_CASE  ) else timestep
         snake_case__     				=					self._index_counter[timestep_int]

        return indices[pos].item()

       @property
       def 		_SCREAMING_SNAKE_CASE(      self  ):






        '''simple docstring'''

        if self.config.timestep_spacing in ["linspace", "trailing"]:
         return self.sigmas.max()

        return (self.sigmas.max() ** 2 + 1) ** 0.5

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       ):






        '''simple docstring'''

        snake_case__     				=					self.index_for_timestep(__SCREAMING_SNAKE_CASE  )

        if self.state_in_first_order:
         snake_case__     				=					self.sigmas[step_index]
        else:
         snake_case__     				=					self.sigmas_interpol[step_index - 1]

        snake_case__     				=					sample / ((sigma**2 + 1) ** 0.5)
        return sample

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE = None ,       __SCREAMING_SNAKE_CASE = None ,       ):






        '''simple docstring'''

        snake_case__     				=					num_inference_steps

        snake_case__     				=					num_train_timesteps or self.config.num_train_timesteps

        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
        if self.config.timestep_spacing == "linspace":
         snake_case__     				=					np.linspace(0 ,       num_train_timesteps - 1 ,       __SCREAMING_SNAKE_CASE ,       dtype=__SCREAMING_SNAKE_CASE  )[::-1].copy()
        elif self.config.timestep_spacing == "leading":
         snake_case__     				=					num_train_timesteps // self.num_inference_steps
         # creates integer timesteps by multiplying by ratio
         # casting to int to avoid issues when num_inference_step is power of 3
         snake_case__     				=					(np.arange(0 ,       __SCREAMING_SNAKE_CASE  ) * step_ratio).round()[::-1].copy().astype(__SCREAMING_SNAKE_CASE  )
         timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
         snake_case__     				=					num_train_timesteps / self.num_inference_steps
         # creates integer timesteps by multiplying by ratio
         # casting to int to avoid issues when num_inference_step is power of 3
         snake_case__     				=					(np.arange(__SCREAMING_SNAKE_CASE ,       0 ,       -step_ratio  )).round().copy().astype(__SCREAMING_SNAKE_CASE  )
         timesteps -= 1
        else:
         raise ValueError(
             F"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."  )

        snake_case__     				=					np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5  )
        snake_case__     				=					torch.from_numpy(np.log(__SCREAMING_SNAKE_CASE  )  ).to(__SCREAMING_SNAKE_CASE  )

        snake_case__     				=					np.interp(__SCREAMING_SNAKE_CASE ,       np.arange(0 ,       len(__SCREAMING_SNAKE_CASE  )  ) ,       __SCREAMING_SNAKE_CASE  )
        snake_case__     				=					np.concatenate([sigmas, [0.0]]  ).astype(np.floataa  )
        snake_case__     				=					torch.from_numpy(__SCREAMING_SNAKE_CASE  ).to(device=__SCREAMING_SNAKE_CASE  )

        # compute up and down sigmas
        snake_case__     				=					sigmas.roll(-1  )
        snake_case__     				=					0.0
        snake_case__     				=					(sigmas_next**2 * (sigmas**2 - sigmas_next**2) / sigmas**2) ** 0.5
        snake_case__     				=					(sigmas_next**2 - sigmas_up**2) ** 0.5
        snake_case__     				=					0.0

        # compute interpolated sigmas
        snake_case__     				=					sigmas.log().lerp(sigmas_down.log() ,       0.5  ).exp()
        snake_case__     				=					0.0

        # set sigmas
        snake_case__     				=					torch.cat([sigmas[:1], sigmas[1:].repeat_interleave(2  ), sigmas[-1:]]  )
        snake_case__     				=					torch.cat(
            [sigmas_interpol[:1], sigmas_interpol[1:].repeat_interleave(2  ), sigmas_interpol[-1:]]  )
        snake_case__     				=					torch.cat([sigmas_up[:1], sigmas_up[1:].repeat_interleave(2  ), sigmas_up[-1:]]  )
        snake_case__     				=					torch.cat([sigmas_down[:1], sigmas_down[1:].repeat_interleave(2  ), sigmas_down[-1:]]  )

        if str(__SCREAMING_SNAKE_CASE  ).startswith("mps"  ):
         # mps does not support float64
         snake_case__     				=					torch.from_numpy(__SCREAMING_SNAKE_CASE  ).to(__SCREAMING_SNAKE_CASE ,       dtype=torch.floataa  )
        else:
         snake_case__     				=					torch.from_numpy(__SCREAMING_SNAKE_CASE  ).to(__SCREAMING_SNAKE_CASE  )

        snake_case__     				=					self.sigma_to_t(__SCREAMING_SNAKE_CASE  ).to(__SCREAMING_SNAKE_CASE ,       dtype=timesteps.dtype  )
        snake_case__     				=					torch.stack((timesteps_interpol[:-2, None], timesteps[1:, None]) ,       dim=-1  ).flatten()

        snake_case__     				=					torch.cat([timesteps[:1], interleaved_timesteps]  )

        snake_case__     				=					None

        # for exp beta schedules, such as the one for `pipeline_shap_e.py`
        # we need an index counter
        snake_case__     				=					defaultdict(__SCREAMING_SNAKE_CASE  )

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE  ):






        '''simple docstring'''

        snake_case__     				=					sigma.log()

        # get distribution
        snake_case__     				=					log_sigma - self.log_sigmas[:, None]

        # get sigmas range
        snake_case__     				=					dists.ge(0  ).cumsum(dim=0  ).argmax(dim=0  ).clamp(max=self.log_sigmas.shape[0] - 2  )
        snake_case__     				=					low_idx + 1

        snake_case__     				=					self.log_sigmas[low_idx]
        snake_case__     				=					self.log_sigmas[high_idx]

        # interpolate sigmas
        snake_case__     				=					(low - log_sigma) / (low - high)
        snake_case__     				=					w.clamp(0 ,       1  )

        # transform interpolation to time range
        snake_case__     				=					(1 - w) * low_idx + w * high_idx
        snake_case__     				=					t.view(sigma.shape  )
        return t

       @property
       def 		_SCREAMING_SNAKE_CASE(      self  ):






        '''simple docstring'''

        return self.sample is None

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE = None ,       __SCREAMING_SNAKE_CASE = True ,       ):






        '''simple docstring'''

        snake_case__     				=					self.index_for_timestep(__SCREAMING_SNAKE_CASE  )

        # advance index counter by 1
        snake_case__     				=					timestep.cpu().item() if torch.is_tensor(__SCREAMING_SNAKE_CASE  ) else timestep
        self._index_counter[timestep_int] += 1

        if self.state_in_first_order:
         snake_case__     				=					self.sigmas[step_index]
         snake_case__     				=					self.sigmas_interpol[step_index]
         snake_case__     				=					self.sigmas_up[step_index]
         snake_case__     				=					self.sigmas_down[step_index - 1]
        else:
         # 2nd order / KPDM2's method
         snake_case__     				=					self.sigmas[step_index - 1]
         snake_case__     				=					self.sigmas_interpol[step_index - 1]
         snake_case__     				=					self.sigmas_up[step_index - 1]
         snake_case__     				=					self.sigmas_down[step_index - 1]

        # currently only gamma=0 is supported. This usually works best anyways.
        # We can support gamma in the future but then need to scale the timestep before
        # passing it to the model which requires a change in API
        snake_case__     				=					0
        snake_case__     				=					sigma * (gamma + 1)  # Note: sigma_hat == sigma for now

        snake_case__     				=					model_output.device
        snake_case__     				=					randn_tensor(model_output.shape ,       dtype=model_output.dtype ,       device=__SCREAMING_SNAKE_CASE ,       generator=__SCREAMING_SNAKE_CASE  )

        # 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
        if self.config.prediction_type == "epsilon":
         snake_case__     				=					sigma_hat if self.state_in_first_order else sigma_interpol
         snake_case__     				=					sample - sigma_input * model_output
        elif self.config.prediction_type == "v_prediction":
         snake_case__     				=					sigma_hat if self.state_in_first_order else sigma_interpol
         snake_case__     				=					model_output * (-sigma_input / (sigma_input**2 + 1) ** 0.5) + (
             sample / (sigma_input**2 + 1)
         )
        elif self.config.prediction_type == "sample":
         raise NotImplementedError("prediction_type not implemented yet: sample"  )
        else:
         raise ValueError(
             F"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"  )

        if self.state_in_first_order:
         # 2. Convert to an ODE derivative for 1st order
         snake_case__     				=					(sample - pred_original_sample) / sigma_hat
         # 3. delta timestep
         snake_case__     				=					sigma_interpol - sigma_hat

         # store for 2nd order step
         snake_case__     				=					sample
         snake_case__     				=					dt
         snake_case__     				=					sample + derivative * dt
        else:
         # DPM-Solver-2
         # 2. Convert to an ODE derivative for 2nd order
         snake_case__     				=					(sample - pred_original_sample) / sigma_interpol
         # 3. delta timestep
         snake_case__     				=					sigma_down - sigma_hat

         snake_case__     				=					self.sample
         snake_case__     				=					None

         snake_case__     				=					sample + derivative * dt
         snake_case__     				=					prev_sample + noise * sigma_up

        if not return_dict:
         return (prev_sample,)

        return SchedulerOutput(prev_sample=__SCREAMING_SNAKE_CASE  )

       def 		_SCREAMING_SNAKE_CASE(      self ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE ,       ):






        '''simple docstring'''

        snake_case__     				=					self.sigmas.to(device=original_samples.device ,       dtype=original_samples.dtype  )
        if original_samples.device.type == "mps" and torch.is_floating_point(__SCREAMING_SNAKE_CASE  ):
         # mps does not support float64
         snake_case__     				=					self.timesteps.to(original_samples.device ,       dtype=torch.floataa  )
         snake_case__     				=					timesteps.to(original_samples.device ,       dtype=torch.floataa  )
        else:
         snake_case__     				=					self.timesteps.to(original_samples.device  )
         snake_case__     				=					timesteps.to(original_samples.device  )

        snake_case__     				=					[self.index_for_timestep(__SCREAMING_SNAKE_CASE ,       __SCREAMING_SNAKE_CASE  ) for t in timesteps]

        snake_case__     				=					sigmas[step_indices].flatten()
        while len(sigma.shape  ) < len(original_samples.shape  ):
         snake_case__     				=					sigma.unsqueeze(-1  )

        snake_case__     				=					original_samples + noise * sigma
        return noisy_samples

       def __len__(      self  ):






        '''simple docstring'''

        return self.config.num_train_timesteps

