





import math




def 		lowerCAmelCase_    (  __UpperCAmelCase:			int      )      ->	bool:
    UpperCamelCase__							:				Union[str, Any]          =      math.loga(math.sqrt(4 * positive_integer + 1      ) / 2 + 1 / 2      )

    return exponent == int(__UpperCAmelCase      )




def 		lowerCAmelCase_    (  __UpperCAmelCase:			float = 1 / 1_2345      )      ->	int:
    UpperCamelCase__							:				Any          =      0
    UpperCamelCase__							:				Optional[int]          =      0

    UpperCamelCase__							:				Dict          =      3
    while True:
        UpperCamelCase__							:				List[str]          =      (integer**2 - 1) / 4
        # if candidate is an integer, then there is a partition for k
        if partition_candidate == int(__UpperCAmelCase      ):
            UpperCamelCase__							:				Dict          =      int(__UpperCAmelCase      )
            total_partitions += 1
            if check_partition_perfect(__UpperCAmelCase      ):
                perfect_partitions += 1
        if perfect_partitions > 0:
            if perfect_partitions / total_partitions < max_proportion:
                return int(__UpperCAmelCase      )
        integer += 1


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








UpperCAmelCase_     							= 'Input must be a string of 8 numbers plus letter'
UpperCAmelCase_     							= 'TRWAGMYFPDXBNJZSQVHLCKE'




def 		lowerCAmelCase_    (  __UpperCAmelCase:			str      )      ->	bool:

    if not isinstance(__UpperCAmelCase		,   __UpperCAmelCase      ):
        UpperCamelCase__							:				Any          =      f"Expected string as input, found {type(__UpperCAmelCase      ).__name__}"
        raise TypeError(__UpperCAmelCase      )

    UpperCamelCase__							:				int          =      spanish_id.replace('''-'''		,   ''''''      ).upper()
    if len(__UpperCAmelCase      ) != 9:
        raise ValueError(__UpperCAmelCase      )

    try:
        UpperCamelCase__							:				List[str]          =      int(spanish_id_clean[0:8]      )
        UpperCamelCase__							:				Optional[int]          =      spanish_id_clean[8]
    except ValueError as ex:
        raise ValueError(__UpperCAmelCase      ) from ex

    if letter.isdigit():
        raise ValueError(__UpperCAmelCase      )

    return letter == LOOKUP_LETTERS[number % 23]


if __name__ == "__main__":
  import doctest

  doctest.testmod()

