



import math





def 		a__      (		_UpperCamelCase      :			int				):
   __lowerCamelCase    				=     math.loga(math.sqrt(4 * positive_integer + 1				) / 2 + 1 / 2				)

   return exponent == int(_UpperCamelCase				)





def 		a__      (		_UpperCamelCase      :			float = 1 / 1_23_45				):
   __lowerCamelCase    				=     0
   __lowerCamelCase    				=     0

   __lowerCamelCase    				=     3
   while True:
      __lowerCamelCase    				=     (integer**2 - 1) / 4
      # if candidate is an integer, then there is a partition for k
      if partition_candidate == int(_UpperCamelCase				):
         __lowerCamelCase    				=     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() = }")











def 		a__      (		_UpperCamelCase      :			int				):
   __lowerCamelCase    				=     int(_UpperCamelCase				)
   if decimal in (0, 1):  # Exit cases for the recursion
      return str(_UpperCamelCase				)
   __lowerCamelCase				,__lowerCamelCase    				=     divmod(_UpperCamelCase			,2				)
   return binary_recursive(_UpperCamelCase				) + str(_UpperCamelCase				)





def 		a__      (		_UpperCamelCase      :			str				):
   __lowerCamelCase    				=     str(_UpperCamelCase				).strip()
   if not number:
      raise ValueError('''No input value was provided'''				)
   __lowerCamelCase    				=     '''-''' if number.startswith('''-'''				) else ''''''
   __lowerCamelCase    				=     number.lstrip('''-'''				)
   if not number.isnumeric():
      raise ValueError('''Input value is not an integer'''				)
   return F"""{negative}0b{binary_recursive(int(_UpperCamelCase				)				)}"""


if __name__ == "__main__":
  from doctest import testmod

  testmod()






