





def      _lowercase					(						_UpperCAmelCase				)						->  Optional[int]:
				lowerCamelCase	 =0
				lowerCamelCase	 =len(_UpperCAmelCase				)

				for i in range(n - 1				):
								for j in range(i + 1			,      _UpperCAmelCase				):
												if arr[i] > arr[j]:
																num_inversions += 1

				return num_inversions





def      _lowercase					(						_UpperCAmelCase				)						->  Union[str, Any]:
				if len(_UpperCAmelCase				) <= 1:
								return arr, 0
				lowerCamelCase	 =len(_UpperCAmelCase				) // 2
				lowerCamelCase	 =arr[0:mid]
				lowerCamelCase	 =arr[mid:]

				lowerCamelCase ,  lowerCamelCase	 =count_inversions_recursive(_UpperCAmelCase				)
				lowerCamelCase ,  lowerCamelCase	 =count_inversions_recursive(_UpperCAmelCase				)
				lowerCamelCase ,  lowerCamelCase	 =_count_cross_inversions(_UpperCAmelCase			,      _UpperCAmelCase				)

				lowerCamelCase	 =inversion_p + inversions_q + cross_inversions
				return c, num_inversions





def      _lowercase					(						_UpperCAmelCase			,      _UpperCAmelCase				)						->  Optional[int]:
				lowerCamelCase	 =[]
				lowerCamelCase	 =lowerCamelCase	 =lowerCamelCase	 =0
				while i < len(_UpperCAmelCase				) and j < len(_UpperCAmelCase				):
								if p[i] > q[j]:
												# if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
												# These are all inversions. The claim emerges from the
												# property that P is sorted.
												num_inversion += len(_UpperCAmelCase				) - i
												r.append(q[j]				)
												j += 1
								else:
												r.append(p[i]				)
												i += 1

				if i < len(_UpperCAmelCase				):
								r.extend(p[i:]				)
				else:
								r.extend(q[j:]				)

				return r, num_inversion





def      _lowercase					(						)						->  int:
				lowerCamelCase	 =[10, 2, 1, 5, 5, 2, 11]

				# this arr has 8 inversions:
				# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

				lowerCamelCase	 =count_inversions_bf(_UpperCAmelCase				)
				lowerCamelCase ,  lowerCamelCase	 =count_inversions_recursive(_UpperCAmelCase				)

				assert num_inversions_bf == num_inversions_recursive == 8

				print("""number of inversions = """			,      _UpperCAmelCase				)

				# testing an array with zero inversion (a sorted arr_1)

				arr_a.sort()
				lowerCamelCase	 =count_inversions_bf(_UpperCAmelCase				)
				lowerCamelCase ,  lowerCamelCase	 =count_inversions_recursive(_UpperCAmelCase				)

				assert num_inversions_bf == num_inversions_recursive == 0
				print("""number of inversions = """			,      _UpperCAmelCase				)

				# an empty list should also have zero inversions
				lowerCamelCase	 =[]
				lowerCamelCase	 =count_inversions_bf(_UpperCAmelCase				)
				lowerCamelCase ,  lowerCamelCase	 =count_inversions_recursive(_UpperCAmelCase				)

				assert num_inversions_bf == num_inversions_recursive == 0
				print("""number of inversions = """			,      _UpperCAmelCase				)


if __name__ == "__main__":
							main()












import math
from collections.abc import Iterator
from itertools import takewhile





def      _lowercase					(						_UpperCAmelCase				)						->  bool:

				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(_UpperCAmelCase				) + 1				)			,      6				):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True





def      _lowercase					(						)						->  Iterator[int]:
				lowerCamelCase	 =2
				while True:
								if is_prime(_UpperCAmelCase				):
												yield num
								num += 1





def      _lowercase					(						_UpperCAmelCase = 2_00_00_00				)						->  int:
				return sum(takewhile(lambda _UpperCAmelCase				: x < n			,      prime_generator()				)				)


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





