




def      _a				(						SCREAMING_SNAKE_CASE_							:    int					):

					# bit count represents no. of bits in the gray code
					if bit_count < 0:
										raise ValueError("The given input must be positive"					)

					# get the generated string sequence
					__lowerCAmelCase           =	gray_code_sequence_string(SCREAMING_SNAKE_CASE_					)
					#
					# convert them to integers
					for i in range(len(SCREAMING_SNAKE_CASE_					)					):
										__lowerCAmelCase           =	int(sequence[i]				,				2					)

					return sequence







def      _a				(						SCREAMING_SNAKE_CASE_							:    int					):

					# The approach is a recursive one
					# Base case achieved when either n = 0 or n=1
					if bit_count == 0:
										return ["0"]

					if bit_count == 1:
										return ["0", "1"]

					__lowerCAmelCase           =	1 << bit_count  # defines the length of the sequence
					# 1<< n is equivalent to 2^n

					# recursive answer will generate answer for n-1 bits
					__lowerCAmelCase           =	gray_code_sequence_string(bit_count - 1					)

					__lowerCAmelCase           =	[]

					# append 0 to first half of the smaller sequence generated
					for i in range(seq_len // 2					):
										__lowerCAmelCase           =	"0" + smaller_sequence[i]
										sequence.append(SCREAMING_SNAKE_CASE_					)

					# append 1 to second half ... start from the end of the list
					for i in reversed(range(seq_len // 2					)					):
										__lowerCAmelCase           =	"1" + smaller_sequence[i]
										sequence.append(SCREAMING_SNAKE_CASE_					)

					return sequence


if __name__ == "__main__":
					import doctest

					doctest.testmod()










import math







def      _a				(						SCREAMING_SNAKE_CASE_							:    int					):

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or number % 2 == 0 or number % 3 == 0:
										# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
										return False

					# All primes number are in format of 6k +/- 1
					for i in range(5				,				int(math.sqrt(SCREAMING_SNAKE_CASE_					) + 1					)				,				6					):
										if number % i == 0 or number % (i + 2) == 0:
															return False
					return True







def      _a				(						SCREAMING_SNAKE_CASE_							:    float = 0.1					):
					__lowerCAmelCase           =	3
					__lowerCAmelCase           =	3

					while primes / (2 * j - 1) >= ratio:
										for i in range(j * j + j + 1				,				(j + 2) * (j + 2)				,				j + 1					):
															primes += is_prime(SCREAMING_SNAKE_CASE_					)
										j += 2
					return j


if __name__ == "__main__":
					import doctest

					doctest.testmod()




