


import math



def     __lowerCAmelCase						(							snake_case	:    int )       ->							bool:
   __lowerCamelCase:				Dict     =	math.loga(math.sqrt(4 * positive_integer + 1 ) / 2 + 1 / 2 )

   return exponent == int(snake_case )



def     __lowerCAmelCase						(							snake_case	:    float = 1 / 12345 )       ->							int:
   __lowerCamelCase:				str     =	0
   __lowerCamelCase:				Optional[Any]     =	0

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


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






def     __lowerCAmelCase						(							snake_case	:    int = 100 )       ->							int:
   __lowerCamelCase:				List[Any]     =	0
   __lowerCamelCase:				int     =	0
   for i in range(1						,      n + 1 ):
      sum_of_squares += i**2
      sum_of_ints += i
   return sum_of_ints**2 - sum_of_squares


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


