import math







def     _a				(      lowercase__  :			int ):







     '''simple docstring'''


     SCREAMING_SNAKE_CASE__				:				str			=		math.loga(math.sqrt(4 * positive_integer + 1 ) / 2 + 1 / 2 )

     return exponent == int(lowercase__ )







def     _a				(      lowercase__  :			float = 1 / 1_23_45 ):







     '''simple docstring'''


     SCREAMING_SNAKE_CASE__				:				List[Any]			=		0
     SCREAMING_SNAKE_CASE__				:				Tuple			=		0

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


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




def     _a				(      lowercase__  :			int = 1_00_00_00 ):







     '''simple docstring'''


     SCREAMING_SNAKE_CASE__				:				str			=		[i - 1 for i in range(limit + 1 )]

     for i in range(2     ,  limit + 1 ):
          if phi[i] == i - 1:
               for j in range(2 * i     ,  limit + 1     ,  lowercase__ ):
                    phi[j] -= phi[j] // i

     return sum(phi[2 : limit + 1] )


if __name__ == "__main__":
      print(solution())



