





'''simple docstring'''




import math






def 	_lowerCAmelCase      (  __snake_case  :			int      )					->					int:

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

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

          __A						:			Union[str, Any]								=	[3, 5]
          __A						:			List[Any]								=	2
          __A						:			Optional[Any]								=	3
          for block in range(1			,							__snake_case      ):
               for _ in range(__snake_case      ):
                    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__   :				str      =   0
    try:
      lowercase__   :				List[str]      =   proth(number)
    except ValueError:
      print(f"""ValueError: there is no {number}th Proth number""")
      continue

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






'''simple docstring'''




def 	_lowerCAmelCase      (  __snake_case  :			str			,							__snake_case  :			str      )					->					float:

     def get_matched_characters(__snake_case  :			str			,							__snake_case  :			str      ) -> str:
          __A						:			Optional[int]								=	[]
          __A						:			Optional[Any]								=	min(len(_stra      )			,							len(_stra      )      ) // 2
          for i, l in enumerate(_stra      ):
               __A						:			str								=	int(max(0			,							i - limit      )      )
               __A						:			str								=	int(min(i + limit + 1			,							len(_stra      )      )      )
               if l in _stra[left:right]:
                    matched.append(__snake_case      )
                    __A						:			Dict								=	f'{_stra[0:_stra.index(__snake_case      )]} {_stra[_stra.index(__snake_case      ) + 1:]}'

          return "".join(__snake_case      )

     # matching characters
     __A						:			Any								=	get_matched_characters(__snake_case			,							__snake_case      )
     __A						:			str								=	get_matched_characters(__snake_case			,							__snake_case      )
     __A						:			str								=	len(__snake_case      )

     # transposition
     __A						:			Tuple								=	(
         len([(ca, ca) for ca, ca in zip(__snake_case			,							__snake_case      ) if ca != ca]      ) // 2
     )

     if not match_count:
          __A						:			int								=	0.0
     else:
          __A						:			Union[str, Any]								=	(
              1
              / 3
              * (
                  match_count / len(__snake_case      )
                  + match_count / len(__snake_case      )
                  + (match_count - transpositions) / match_count
              )
          )

     # common prefix up to 4 characters
     __A						:			Tuple								=	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'''))