



import math






def 				A				(					_lowerCamelCase  ):





							'''simple docstring'''




							_lowerCAmelCase     :							Optional[Any]    =    [True] * n
							_lowerCAmelCase     :							Union[str, Any]    =    False
							_lowerCAmelCase     :							Optional[Any]    =    False
							_lowerCAmelCase     :							Tuple    =    True

							for i in range(3				, int(n**0.5 + 1  )				, 2  ):
														_lowerCAmelCase     :							str    =    i * 2
														while index < n:
																					_lowerCAmelCase     :							List[str]    =    False
																					_lowerCAmelCase     :							Optional[int]    =    index + i

							_lowerCAmelCase     :							Any    =    [2]

							for i in range(3				, _lowerCamelCase				, 2  ):
														if is_prime[i]:
																					primes.append(_lowerCamelCase  )

							return primes






def 				A				(					_lowerCamelCase = 999_966_663_333  ):





							'''simple docstring'''




							_lowerCAmelCase     :							Optional[int]    =    math.floor(math.sqrt(_lowerCamelCase  )  ) + 100
							_lowerCAmelCase     :							str    =    prime_sieve(_lowerCamelCase  )

							_lowerCAmelCase     :							Optional[int]    =    0
							_lowerCAmelCase     :							List[str]    =    0
							_lowerCAmelCase     :							List[Any]    =    primes[prime_index]

							while (last_prime**2) <= limit:
														_lowerCAmelCase     :							Tuple    =    primes[prime_index + 1]

														_lowerCAmelCase     :							List[Any]    =    last_prime**2
														_lowerCAmelCase     :							Dict    =    next_prime**2

														# Get numbers divisible by lps(current)
														_lowerCAmelCase     :							List[Any]    =    lower_bound + last_prime
														while upper_bound > current <= limit:
																					matches_sum += current
																					current += last_prime

														# Reset the upper_bound
														while (upper_bound - next_prime) > limit:
																					upper_bound -= next_prime

														# Add the numbers divisible by ups(current)
														_lowerCAmelCase     :							List[Any]    =    upper_bound - next_prime
														while current > lower_bound:
																					matches_sum += current
																					current -= next_prime

														# Remove the numbers divisible by both ups and lps
														_lowerCAmelCase     :							Optional[int]    =    0
														while upper_bound > current <= limit:
																					if current <= lower_bound:
																												# Increment the current number
																												current += last_prime * next_prime
																												continue

																					if current > limit:
																												break

																					# Remove twice since it was added by both ups and lps
																					matches_sum -= current * 2

																					# Increment the current number
																					current += last_prime * next_prime

														# Setup for next pair
														_lowerCAmelCase     :							Optional[int]    =    next_prime
														prime_index += 1

							return matches_sum


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





def 				A				(					_lowerCamelCase  ):





							'''simple docstring'''




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

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

							return sequence






def 				A				(					_lowerCamelCase  ):





							'''simple docstring'''




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

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

							_lowerCAmelCase     :							List[Any]    =    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     :							Optional[int]    =    gray_code_sequence_string(bit_count - 1  )

							_lowerCAmelCase     :							str    =    []

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

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

							return sequence


if __name__ == "__main__":
						import doctest

						doctest.testmod()
