






"""simple docstring"""







def _a			(		_snake_case							):







						"""simple docstring"""






						UpperCAmelCase          =	0
						UpperCAmelCase          =	len(_snake_case							)

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

						return num_inversions

def _a			(		_snake_case							):







						"""simple docstring"""






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

						UpperCAmelCase      , UpperCAmelCase          =	count_inversions_recursive(_snake_case							)
						UpperCAmelCase      , UpperCAmelCase          =	count_inversions_recursive(_snake_case							)
						UpperCAmelCase      , UpperCAmelCase          =	_count_cross_inversions(_snake_case				,	_snake_case							)

						UpperCAmelCase          =	inversion_p + inversions_q + cross_inversions
						return c, num_inversions

def _a			(		_snake_case				,	_snake_case							):







						"""simple docstring"""






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

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

						return r, num_inversion

def _a			(		):







						"""simple docstring"""






						UpperCAmelCase          =	[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)

						UpperCAmelCase          =	count_inversions_bf(_snake_case							)
						UpperCAmelCase      , UpperCAmelCase          =	count_inversions_recursive(_snake_case							)

						assert num_inversions_bf == num_inversions_recursive == 8

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

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

						arr_a.sort()
						UpperCAmelCase          =	count_inversions_bf(_snake_case							)
						UpperCAmelCase      , UpperCAmelCase          =	count_inversions_recursive(_snake_case							)

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

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

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


if __name__ == "__main__":
		main()











"""simple docstring"""







def _a			(		_snake_case = 6008_5147_5143							):







      """simple docstring"""






      try:
            UpperCAmelCase          =	int(_snake_case							)
      except (TypeError, ValueError):
            raise TypeError("""Parameter n must be int or castable to int."""							)
      if n <= 0:
            raise ValueError("""Parameter n must be greater than or equal to one."""							)
      UpperCAmelCase          =	1
      UpperCAmelCase          =	2
      while i * i <= n:
            while n % i == 0:
                  UpperCAmelCase          =	i
                  n //= i
            i += 1
      if n > 1:
            UpperCAmelCase          =	n
      return int(_snake_case							)


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



