


from __future__ import annotations

from functools import lru_cache
from math import ceil

_snake_case					:		Tuple      =    100

_snake_case					:		int      =    set(range(3, NUM_PRIMES, 2))
primes.add(2)
_snake_case					:		int

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
			if prime not in primes:
						continue
			primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))







@lru_cache(maxsize=1_0_0     )
def     lowerCAmelCase_					( __lowerCamelCase     ):
							if number_to_partition < 0:
														return set()
							elif number_to_partition == 0:
														return {1}

							__snake_case		:      set[int]										=       set()
							__snake_case		:      int
							__snake_case		:      int

							for prime in primes:
														if prime > number_to_partition:
																					continue
														for sub in partition(number_to_partition - prime     ):
																					ret.add(sub * prime     )

							return ret







def     lowerCAmelCase_					( __lowerCamelCase = 5_0_0_0     ):
							for number_to_partition in range(1							,     __lowerCamelCase     ):
														if len(partition(__lowerCamelCase     )     ) > number_unique_partitions:
																					return number_to_partition
							return None


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





import importlib
import math
import os
from dataclasses import dataclass
from enum import Enum
from typing import Any, Dict, Optional, Tuple, Union

import flax
import jax.numpy as jnp

from ..utils import BaseOutput


_snake_case					:		int      =    "scheduler_config.json"

class     a		(_lowerCAmelCase				):




							"""simple docstring"""


							__UpperCAmelCase   :							Tuple					  =  1
							__UpperCAmelCase   :							Tuple					  =  2
							__UpperCAmelCase   :							Union[str, Any]					  =  3
							__UpperCAmelCase   :							List[Any]					  =  4
							__UpperCAmelCase   :							Tuple					  =  5

@dataclass
class     a		(_lowerCAmelCase				):




							"""simple docstring"""


							__UpperCAmelCase   :							jnp.ndarray

class     a		:




							"""simple docstring"""


							__UpperCAmelCase   :							Dict					  =  SCHEDULER_CONFIG_NAME
							__UpperCAmelCase   :							Union[str, Any]					  =  ["dtype"]
							__UpperCAmelCase   :							Tuple					  =  []
							__UpperCAmelCase   :							int					  =  True

							@classmethod
							def    __snake_case (							cls	:  List[str]   ,     lowerCamelCase	:  Dict[str, Any] = None   ,     lowerCamelCase	:  Optional[str] = None   ,     lowerCamelCase	:  List[str]=False   ,     **lowerCamelCase	:  Union[str, Any]   ,     )     ->   List[str]:
														__snake_case			,      __snake_case		:      List[str]										=       cls.load_config(
														    pretrained_model_name_or_path=lowerCamelCase   ,     subfolder=lowerCamelCase   ,     return_unused_kwargs=lowerCamelCase   ,     **lowerCamelCase   ,     )
														__snake_case			,      __snake_case		:      Dict										=       cls.from_config(lowerCamelCase   ,     return_unused_kwargs=lowerCamelCase   ,     **lowerCamelCase    )

														if hasattr(lowerCamelCase   ,     "create_state"    ) and getattr(lowerCamelCase   ,     "has_state"   ,     lowerCamelCase    ):
																					__snake_case		:      Tuple										=       scheduler.create_state()

														if return_unused_kwargs:
																					return scheduler, state, unused_kwargs

														return scheduler, state

							def    __snake_case (							self	:  Any   ,     lowerCamelCase	:  Union[str, os.PathLike]   ,     lowerCamelCase	:  bool = False   ,     **lowerCamelCase	:  List[Any]    )     ->   int:
														self.save_config(save_directory=lowerCamelCase   ,     push_to_hub=lowerCamelCase   ,     **lowerCamelCase    )

							@property
							def    __snake_case (							self	:  Tuple    )     ->   List[Any]:
														return self._get_compatibles()






							@classmethod
							def    __snake_case (							cls	:  int    )     ->   Dict:
														__snake_case		:      Tuple										=       list(set([cls.__name__] + cls._compatibles    )    )
														__snake_case		:      int										=       importlib.import_module(__name__.split("."    )[0]    )
														__snake_case		:      Tuple										=       [
														    getattr(lowerCamelCase   ,     lowerCamelCase    ) for c in compatible_classes_str if hasattr(lowerCamelCase   ,     lowerCamelCase    )
														]
														return compatible_classes







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase     ):
							assert len(__lowerCamelCase     ) >= x.ndim
							return jnp.broadcast_to(x.reshape(x.shape + (1,) * (len(__lowerCamelCase     ) - x.ndim)     )							,     __lowerCamelCase     )







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase=0.9_9_9							,     __lowerCamelCase=jnp.floataa     ):

							def alpha_bar(__lowerCamelCase     ):
														return math.cos((time_step + 0.0_0_8) / 1.0_0_8 * math.pi / 2     ) ** 2

							__snake_case		:      List[Any]										=       []
							for i in range(__lowerCamelCase     ):
														__snake_case		:      Dict										=       i / num_diffusion_timesteps
														__snake_case		:      str										=       (i + 1) / num_diffusion_timesteps
														betas.append(min(1 - alpha_bar(__lowerCamelCase     ) / alpha_bar(__lowerCamelCase     )							,     __lowerCamelCase     )     )
							return jnp.array(__lowerCamelCase							,     dtype=__lowerCamelCase     )

@flax.struct.dataclass
class     a		:




							"""simple docstring"""


							__UpperCAmelCase   :							jnp.ndarray
							__UpperCAmelCase   :							jnp.ndarray
							__UpperCAmelCase   :							jnp.ndarray

							@classmethod
							def    __snake_case (							cls	:  Union[str, Any]   ,     lowerCamelCase	:  int    )     ->   List[Any]:
														__snake_case		:      Dict										=       scheduler.config

														if config.trained_betas is not None:
																					__snake_case		:      Dict										=       jnp.asarray(config.trained_betas   ,     dtype=scheduler.dtype    )
														elif config.beta_schedule == "linear":
																					__snake_case		:      Optional[int]										=       jnp.linspace(config.beta_start   ,     config.beta_end   ,     config.num_train_timesteps   ,     dtype=scheduler.dtype    )
														elif config.beta_schedule == "scaled_linear":
																					# this schedule is very specific to the latent diffusion model.
																					__snake_case		:      str										=       (
																					    jnp.linspace(
																					        config.beta_start**0.5   ,     config.beta_end**0.5   ,     config.num_train_timesteps   ,     dtype=scheduler.dtype    )
																					    ** 2
																					)
														elif config.beta_schedule == "squaredcos_cap_v2":
																					# Glide cosine schedule
																					__snake_case		:      Optional[Any]										=       betas_for_alpha_bar(config.num_train_timesteps   ,     dtype=scheduler.dtype    )
														else:
																					raise NotImplementedError(
																					    F'beta_schedule {config.beta_schedule} is not implemented for scheduler {scheduler.__class__.__name__}'    )

														__snake_case		:      Any										=       1.0 - betas

														__snake_case		:      int										=       jnp.cumprod(lowerCamelCase   ,     axis=0    )

														return cls(
														    alphas=lowerCamelCase   ,     betas=lowerCamelCase   ,     alphas_cumprod=lowerCamelCase   ,     )







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     ):
							__snake_case		:      List[Any]										=       state.alphas_cumprod

							__snake_case		:      str										=       alphas_cumprod[timesteps] ** 0.5
							__snake_case		:      Dict										=       sqrt_alpha_prod.flatten()
							__snake_case		:      str										=       broadcast_to_shape_from_left(__lowerCamelCase							,     original_samples.shape     )

							__snake_case		:      Tuple										=       (1 - alphas_cumprod[timesteps]) ** 0.5
							__snake_case		:      str										=       sqrt_one_minus_alpha_prod.flatten()
							__snake_case		:      Tuple										=       broadcast_to_shape_from_left(__lowerCamelCase							,     original_samples.shape     )

							return sqrt_alpha_prod, sqrt_one_minus_alpha_prod







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     ):
							__snake_case			,      __snake_case		:      Union[str, Any]										=       get_sqrt_alpha_prod(__lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     )
							__snake_case		:      List[str]										=       sqrt_alpha_prod * original_samples + sqrt_one_minus_alpha_prod * noise
							return noisy_samples







def     lowerCAmelCase_					( __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     ):
							__snake_case			,      __snake_case		:      Dict										=       get_sqrt_alpha_prod(__lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase							,     __lowerCamelCase     )
							__snake_case		:      Optional[int]										=       sqrt_alpha_prod * noise - sqrt_one_minus_alpha_prod * sample
							return velocity

