





import math





def      lowerCAmelCase_ (       __a		)     ->		bool:







  """simple docstring"""





  lowerCamelCase__:      Any			  =math.loga(math.sqrt(4 * positive_integer + 1		) / 2 + 1 / 2		)

  return exponent == int(__a		)





def      lowerCAmelCase_ (       __a = 1 / 12345		)     ->		int:







  """simple docstring"""





  lowerCamelCase__:      Union[str, Any]			  =0
  lowerCamelCase__:      List[Any]			  =0

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


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








from math import ceil, sqrt





def      lowerCAmelCase_ (       __a = 1000000		)     ->		int:







  """simple docstring"""





  lowerCamelCase__:      Optional[int]			  =0

  for outer_width in range(3	,    (limit // 4) + 2		):
    if outer_width**2 > limit:
      lowerCamelCase__:      Dict			  =max(ceil(sqrt(outer_width**2 - limit		)		)	,    1		)
    else:
      lowerCamelCase__:      str			  =1
    if (outer_width - hole_width_lower_bound) % 2:
      hole_width_lower_bound += 1

    answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

  return answer


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

