
def     __lowerCAmelCase			(			a__						,    a__  )       ->  float:

				def get_matched_characters(a__						,    a__  ) -> str:
								__a										=    []
								__a										=    min(len(_stra  )						,    len(_stra  )  ) // 2
								for i, l in enumerate(_stra  ):
												__a										=    int(max(0						,    i - limit  )  )
												__a										=    int(min(i + limit + 1						,    len(_stra  )  )  )
												if l in _stra[left:right]:
																matched.append(a__  )
																__a										=    F"""{_stra[0:_stra.index(a__  )]} {_stra[_stra.index(a__  ) + 1:]}"""

								return "".join(a__  )

				# matching characters
				__a										=    get_matched_characters(a__						,    a__  )
				__a										=    get_matched_characters(a__						,    a__  )
				__a										=    len(a__  )

				# transposition
				__a										=    (
				    len([(ca, ca) for ca, ca in zip(a__						,    a__  ) if ca != ca]  ) // 2
				)

				if not match_count:
								__a										=    0.0
				else:
								__a										=    (
								    1
								    / 3
								    * (
								        match_count / len(a__  )
								        + match_count / len(a__  )
								        + (match_count - transpositions) / match_count
								    )
								)

				# common prefix up to 4 characters
				__a										=    0
				for ca, ca in zip(stra[:4]						,    stra[:4]  ):
								if ca == ca:
												prefix_len += 1
								else:
												break

				return jaro + 0.1 * prefix_len * (1 - jaro)


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					print(jaro_winkler('hello', 'world'))
'''simple docstring'''




import math




def 							lowerCamelCase	(      UpperCAmelCase__       :					int )			->      int:

							if not isinstance(UpperCAmelCase__      ,      UpperCAmelCase__ ):
														lowercase_					:			List[Any]										=       F'''Input value of [number={number}] must be an integer'''
														raise TypeError(UpperCAmelCase__ )

							if number < 1:
														lowercase_					:			Tuple										=       F'''Input value of [number={number}] must be > 0'''
														raise ValueError(UpperCAmelCase__ )
							elif number == 1:
														return 3
							elif number == 2:
														return 5
							else:
														lowercase_					:			Union[str, Any]										=       int(math.log(number // 3      ,      2 ) ) + 2

														lowercase_					:			Any										=       [3, 5]
														lowercase_					:			List[Any]										=       2
														lowercase_					:			Optional[Any]										=       3
														for block in range(1      ,      UpperCAmelCase__ ):
																					for _ in range(UpperCAmelCase__ ):
																												proth_list.append(2 ** (block + 1) + proth_list[proth_index - 1] )
																												proth_index += 1
																					increment *= 2

							return proth_list[number - 1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						for number in range(11):
												_lowercase						: Optional[Any]						=						0
												try:
																		_lowercase						: str						=						proth(number)
												except ValueError:
																		print(f"""ValueError: there is no {number}th Proth number""")
																		continue

												print(f"""The {number}th Proth number: {value}""")





