
def 		UpperCAmelCase     (      a_		)						->      str:







				"""simple docstring"""



				__A       =					0
				__A       =					len(a_		)

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

				return num_inversions


def 		UpperCAmelCase     (      a_		)						->      Optional[int]:







				"""simple docstring"""



				if len(a_		) <= 1:
								return arr, 0
				__A       =					len(a_		) // 2
				__A       =					arr[0:mid]
				__A       =					arr[mid:]

				__A     ,				__A       =					count_inversions_recursive(a_		)
				__A     ,				__A       =					count_inversions_recursive(a_		)
				__A     ,				__A       =					_count_cross_inversions(a_						,  a_		)

				__A       =					inversion_p + inversions_q + cross_inversions
				return c, num_inversions


def 		UpperCAmelCase     (      a_						,  a_		)						->      Union[str, Any]:







				"""simple docstring"""



				__A       =					[]
				__A       =					__A       =					__A       =					0
				while i < len(a_		) and j < len(a_		):
								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(a_		) - i
												r.append(q[j]		)
												j += 1
								else:
												r.append(p[i]		)
												i += 1

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

				return r, num_inversion


def 		UpperCAmelCase     (      )						->      Optional[int]:







				"""simple docstring"""



				__A       =					[1_0, 2, 1, 5, 5, 2, 1_1]

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

				__A       =					count_inversions_bf(a_		)
				__A     ,				__A       =					count_inversions_recursive(a_		)

				assert num_inversions_bf == num_inversions_recursive == 8

				print("number of inversions = "						,  a_		)

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

				arr_a.sort()
				__A       =					count_inversions_bf(a_		)
				__A     ,				__A       =					count_inversions_recursive(a_		)

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

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

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


if __name__ == "__main__":
					main()




"""simple docstring"""



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







def     __lowerCAmelCase		(		lowercase    :    int					)      ->		bool:


    """simple docstring"""





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







def     __lowerCAmelCase		(		)      ->		Iterator[int]:


    """simple docstring"""





    snake_case       : Optional[int]    =       2
    while True:
        if is_prime(lowercase					):
            yield num
        num += 1







def     __lowerCAmelCase		(		lowercase    :    int = 200_0000					)      ->		int:


    """simple docstring"""





    return sum(takewhile(lambda lowercase					: x < n		,				prime_generator()					)					)


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





