






from __future__ import annotations

from functools import lru_cache
from math import ceil

__snake_case    :    Union[str, Any]         =     1_00

__snake_case    :    Tuple         =     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=100		)
def 	_UpperCamelCase   (    UpperCamelCase_       :					int		)  ->   set[int]:

						"""simple docstring"""



						if number_to_partition < 0:
												return set()
						elif number_to_partition == 0:
												return {1}

						lowerCAmelCase__    						=						set()
						lowerCAmelCase__    						=						42
						lowerCAmelCase__    						=						42

						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 	_UpperCamelCase   (    UpperCamelCase_       :					int = 5000		)  ->   int | None:

						"""simple docstring"""



						for number_to_partition in range(1    ,       UpperCamelCase_		):
												if len(partition(UpperCamelCase_		)		) > number_unique_partitions:
																		return number_to_partition
						return None


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













def 	_UpperCamelCase   (    UpperCamelCase_       :					int		)  ->   int:

						"""simple docstring"""



						if not isinstance(UpperCamelCase_    ,       UpperCamelCase_		):
												raise ValueError('multiplicative_persistence() only accepts integral values'		)
						if num < 0:
												raise ValueError('multiplicative_persistence() does not accept negative values'		)

						lowerCAmelCase__    						=						0
						lowerCAmelCase__    						=						str(UpperCamelCase_		)

						while len(UpperCamelCase_		) != 1:
												lowerCAmelCase__    						=						[int(UpperCamelCase_		) for i in num_string]

												lowerCAmelCase__    						=						1
												for i in range(0    ,       len(UpperCamelCase_		)		):
																		total *= numbers[i]

												lowerCAmelCase__    						=						str(UpperCamelCase_		)

												steps += 1
						return steps







def 	_UpperCamelCase   (    UpperCamelCase_       :					int		)  ->   int:

						"""simple docstring"""



						if not isinstance(UpperCamelCase_    ,       UpperCamelCase_		):
												raise ValueError('additive_persistence() only accepts integral values'		)
						if num < 0:
												raise ValueError('additive_persistence() does not accept negative values'		)

						lowerCAmelCase__    						=						0
						lowerCAmelCase__    						=						str(UpperCamelCase_		)

						while len(UpperCamelCase_		) != 1:
												lowerCAmelCase__    						=						[int(UpperCamelCase_		) for i in num_string]

												lowerCAmelCase__    						=						0
												for i in range(0    ,       len(UpperCamelCase_		)		):
																		total += numbers[i]

												lowerCAmelCase__    						=						str(UpperCamelCase_		)

												steps += 1
						return steps


if __name__ == "__main__":
					import doctest

					doctest.testmod()





