





"""simple docstring"""





import math


def       a__					(					_SCREAMING_SNAKE_CASE    ):




		"""simple docstring"""


		UpperCamelCase   =	[True] * n
		UpperCamelCase   =	False
		UpperCamelCase   =	False
		UpperCamelCase   =	True

		for i in range(3  ,	int(n**0.5 + 1    )  ,	2    ):
				UpperCamelCase   =	i * 2
				while index < n:
						UpperCamelCase   =	False
						UpperCamelCase   =	index + i

		UpperCamelCase   =	[2]

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

		return primes


def       a__					(					_SCREAMING_SNAKE_CASE = 999_966_663_333    ):




		"""simple docstring"""


		UpperCamelCase   =	math.floor(math.sqrt(_SCREAMING_SNAKE_CASE    )    ) + 100
		UpperCamelCase   =	prime_sieve(_SCREAMING_SNAKE_CASE    )

		UpperCamelCase   =	0
		UpperCamelCase   =	0
		UpperCamelCase   =	primes[prime_index]

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

				UpperCamelCase   =	last_prime**2
				UpperCamelCase   =	next_prime**2

				# Get numbers divisible by lps(current)
				UpperCamelCase   =	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)
				UpperCamelCase   =	upper_bound - next_prime
				while current > lower_bound:
						matches_sum += current
						current -= next_prime

				# Remove the numbers divisible by both ups and lps
				UpperCamelCase   =	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
				UpperCamelCase   =	next_prime
				prime_index += 1

		return matches_sum


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













"""simple docstring"""





import pytest

from datasets.parallel import ParallelBackendConfig, parallel_backend
from datasets.utils.py_utils import map_nested

from .utils import require_dill_gt_0_3_2, require_joblibspark, require_not_windows


def       a__					(					_SCREAMING_SNAKE_CASE    ):  # picklable for multiprocessing




		"""simple docstring"""


		return i + 1


@require_dill_gt_0_3_2
@require_joblibspark
@require_not_windows
def       a__					(					):




		"""simple docstring"""


		with parallel_backend("spark"    ):
				assert ParallelBackendConfig.backend_name == "spark"

		UpperCamelCase   =	[1, 2, 3]
		with pytest.raises(_SCREAMING_SNAKE_CASE    ):
				with parallel_backend("unsupported backend"    ):
						map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=2    )

		with pytest.raises(_SCREAMING_SNAKE_CASE    ):
				with parallel_backend("unsupported backend"    ):
						map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=-1    )






@require_dill_gt_0_3_2
@require_joblibspark
@require_not_windows
@pytest.mark.parametrize("num_proc"  ,	[2, -1]    )
def       a__					(					_SCREAMING_SNAKE_CASE    ):




		"""simple docstring"""


		UpperCamelCase   =	[1, 2]
		UpperCamelCase   =	{"a": 1, "b": 2}
		UpperCamelCase   =	{"a": [1, 2], "b": [3, 4]}
		UpperCamelCase   =	{"a": {"1": 1}, "b": 2}
		UpperCamelCase   =	{"a": 1, "b": 2, "c": 3, "d": 4}
		UpperCamelCase   =	[2, 3]
		UpperCamelCase   =	{"a": 2, "b": 3}
		UpperCamelCase   =	{"a": [2, 3], "b": [4, 5]}
		UpperCamelCase   =	{"a": {"1": 2}, "b": 3}
		UpperCamelCase   =	{"a": 2, "b": 3, "c": 4, "d": 5}

		with parallel_backend("spark"    ):
				assert map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=_SCREAMING_SNAKE_CASE    ) == expected_map_nested_sa
				assert map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=_SCREAMING_SNAKE_CASE    ) == expected_map_nested_sa
				assert map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=_SCREAMING_SNAKE_CASE    ) == expected_map_nested_sa
				assert map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=_SCREAMING_SNAKE_CASE    ) == expected_map_nested_sa
				assert map_nested(_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	num_proc=_SCREAMING_SNAKE_CASE    ) == expected_map_nested_sa






