





import math






def      lowerCamelCase_						(				_UpperCamelCase )		->						bool:






				"""simple docstring"""


				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5 ,				int(math.sqrt(_UpperCamelCase ) + 1 ) ,				6 ):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True






def      lowerCamelCase_						(				_UpperCamelCase = 10_001 )		->						int:






				"""simple docstring"""


				try:
								snake_case_						:    int				=  int(_UpperCamelCase )
				except (TypeError, ValueError):
								raise TypeError('''Parameter nth must be int or castable to int.''' ) from None
				if nth <= 0:
								raise ValueError('''Parameter nth must be greater than or equal to one.''' )
				snake_case_						:    list[int]				=  []
				snake_case_						:    Tuple				=  2
				while len(_UpperCamelCase ) < nth:
								if is_prime(_UpperCamelCase ):
												primes.append(_UpperCamelCase )
												num += 1
								else:
												num += 1
				return primes[len(_UpperCamelCase ) - 1]


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







from math import factorial

lowerCAmelCase_    =				{str(digit): factorial(digit) for digit in range(1_0)}






def      lowerCamelCase_						(				_UpperCamelCase )		->						int:






				"""simple docstring"""


				if not isinstance(_UpperCamelCase ,				_UpperCamelCase ):
								raise TypeError('''Parameter number must be int''' )

				if number < 0:
								raise ValueError('''Parameter number must be greater than or equal to 0''' )

				# Converts number in string to iterate on its digits and adds its factorial.
				return sum(DIGIT_FACTORIAL[digit] for digit in str(_UpperCamelCase ) )






def      lowerCamelCase_						(				_UpperCamelCase = 60 ,				_UpperCamelCase = 1_000_000 )		->						int:






				"""simple docstring"""


				if not isinstance(_UpperCamelCase ,				_UpperCamelCase ) or not isinstance(_UpperCamelCase ,				_UpperCamelCase ):
								raise TypeError('''Parameters chain_length and number_limit must be int''' )

				if chain_length <= 0 or number_limit <= 0:
								raise ValueError(
								    '''Parameters chain_length and number_limit must be greater than 0''' )

				# the counter for the chains with the exact desired length
				snake_case_						:    Optional[Any]				=  0
				# the cached sizes of the previous chains
				snake_case_						:    dict[int, int]				=  {}

				for start_chain_element in range(1 ,				_UpperCamelCase ):
								# The temporary set will contain the elements of the chain
								snake_case_						:    List[str]				=  set()
								snake_case_						:    List[Any]				=  0

								# Stop computing the chain when you find a cached size, a repeating item or the
								# length is greater then the desired one.
								snake_case_						:    Any				=  start_chain_element
								while (
								    chain_element not in chain_sets_lengths
								    and chain_element not in chain_set
								    and chain_set_length <= chain_length
								):
												chain_set.add(_UpperCamelCase )
												chain_set_length += 1
												snake_case_						:    List[Any]				=  digit_factorial_sum(_UpperCamelCase )

								if chain_element in chain_sets_lengths:
												chain_set_length += chain_sets_lengths[chain_element]

								snake_case_						:    List[str]				=  chain_set_length

								# If chain contains the exact amount of elements increase the counter
								if chain_set_length == chain_length:
												chains_counter += 1

				return chains_counter


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	print(F'''{solution()}''')
