


'''simple docstring'''
def       a				(      __a       )  ->     list:



					'''simple docstring'''






					if bit_count < 0:
										raise ValueError('''The given input must be positive'''       )

					# get the generated string sequence
					UpperCamelCase__       :int							 =			gray_code_sequence_string(__a       )
					#
					# convert them to integers
					for i in range(len(__a       )       ):
										UpperCamelCase__       :Union[str, Any]							 =			int(sequence[i]			, 2       )

					return sequence


def       a				(      __a       )  ->     list:



					'''simple docstring'''






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

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

					UpperCamelCase__       :Tuple							 =			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
					UpperCamelCase__       :Optional[Any]							 =			gray_code_sequence_string(bit_count - 1       )

					UpperCamelCase__       :Tuple							 =			[]

					# append 0 to first half of the smaller sequence generated
					for i in range(seq_len // 2       ):
										UpperCamelCase__       :Any							 =			'''0''' + smaller_sequence[i]
										sequence.append(__a       )

					# append 1 to second half ... start from the end of the list
					for i in reversed(range(seq_len // 2       )       ):
										UpperCamelCase__       :List[Any]							 =			'''1''' + smaller_sequence[i]
										sequence.append(__a       )

					return sequence


if __name__ == "__main__":
		import doctest

		doctest.testmod()



'''simple docstring'''
from __future__ import annotations

__snake_case											=					[True] * 1000001
__snake_case											=					2
while i * i <= 1000000:
		if seive[i]:
				for j in range(i * i, 1000001, i):
						__snake_case											=					False
		i += 1


def       a				(      __a       )  ->     bool:



					'''simple docstring'''






					return seive[n]


def       a				(      __a       )  ->     bool:



					'''simple docstring'''






					return any(digit in '''02468''' for digit in str(__a       )       )


def       a				(      __a = 1000000       )  ->     list[int]:



					'''simple docstring'''






					UpperCamelCase__       :Any							 =			[2]  # result already includes the number 2.
					for num in range(3			, limit + 1			, 2       ):
										if is_prime(__a       ) and not contains_an_even_digit(__a       ):
															UpperCamelCase__       :str							 =			str(__a       )
															UpperCamelCase__       :List[str]							 =			[int(str_num[j:] + str_num[:j]       ) for j in range(len(__a       )       )]
															if all(is_prime(__a       ) for i in list_nums       ):
																				result.append(__a       )
					return result


def       a				(      )  ->     int:



					'''simple docstring'''






					return len(find_circular_primes()       )


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