






import math

def      _lowerCamelCase	(						__A    :							int )      ->					int:

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

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

        _UpperCAmelCase				:      Dict			  =				[3, 5]
        _UpperCAmelCase				:      Tuple			  =				2
        _UpperCAmelCase				:      Dict			  =				3
        for block in range(1		,		__A ):
            for _ in range(__A ):
                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):
    SCREAMING_SNAKE_CASE 							=						0
    try:
      SCREAMING_SNAKE_CASE 							=						proth(number)
    except ValueError:
      print(F'ValueError: there is no {number}th Proth number')
      continue

    print(F'The {number}th Proth number: {value}')












from __future__ import annotations

def      _lowerCamelCase	(						__A    :							int )      ->					list[int]:
    _UpperCAmelCase				:      List[str]			  =				[True] * limit
    _UpperCAmelCase				:      Optional[int]			  =				False
    _UpperCAmelCase				:      Dict			  =				False
    _UpperCAmelCase				:      List[str]			  =				True

    for i in range(3		,		int(limit**0.5 + 1 )		,		2 ):
        _UpperCAmelCase				:      List[str]			  =				i * 2
        while index < limit:
            _UpperCAmelCase				:      Union[str, Any]			  =				False
            _UpperCAmelCase				:      int			  =				index + i

    _UpperCAmelCase				:      Optional[int]			  =				[2]

    for i in range(3		,		__A		,		2 ):
        if is_prime[i]:
            primes.append(__A )

    return primes

def      _lowerCamelCase	(						__A    :							int = 1_000_000 )      ->					int:
    _UpperCAmelCase				:      Any			  =				prime_sieve(__A )
    _UpperCAmelCase				:      Optional[Any]			  =				0
    _UpperCAmelCase				:      Tuple			  =				0

    for i in range(len(__A ) ):
        for j in range(i + length		,		len(__A ) ):
            _UpperCAmelCase				:      List[Any]			  =				sum(primes[i:j] )
            if sol >= ceiling:
                break

            if sol in primes:
                _UpperCAmelCase				:      List[str]			  =				j - i
                _UpperCAmelCase				:      Optional[Any]			  =				sol

    return largest


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




