


'''simple docstring'''



def    lowerCamelCase__   ( A     :    int = 1_00_00_00  ):




						'''simple docstring'''






						UpperCAmelCase         =			limit + 1
						UpperCAmelCase         =			[0] * limit
						for first_term in range(1   ,     A  ):
												for n in range(A   ,     A   ,     A  ):
																		UpperCAmelCase         =			first_term + n / first_term
																		if common_difference % 4:  # d must be divisble by 4
																								continue
																		else:
																								common_difference /= 4
																								if (
																								    first_term > common_difference
																								    and first_term < 4 * common_difference
																								):  # since x,y,z are positive integers
																														frequency[n] += 1  # so z>0 and a>d ,also 4d<a

						UpperCAmelCase         =			sum(1 for x in frequency[1:limit] if x == 10  )

						return count


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









'''simple docstring'''



import heapq


def    lowerCamelCase__   ( A     :    dict  ):




						'''simple docstring'''






						UpperCAmelCase         =			[]

						# for each node and his adjacency list add them and the rank of the node to queue
						# using heapq module the queue will be filled like a Priority Queue
						# heapq works with a min priority queue, so I used -1*len(v) to build it
						for key, value in graph.items():
												# O(log(n))
												heapq.heappush(A   ,     [-1 * len(A  ), (key, value)]  )

						# chosen_vertices = set of chosen vertices
						UpperCAmelCase         =			set()

						# while queue isn't empty and there are still edges
						#   (queue[0][0] is the rank of the node with max rank)
						while queue and queue[0][0] != 0:
												# extract vertex with max rank from queue and add it to chosen_vertices
												UpperCAmelCase         =			heapq.heappop(A  )[1][0]
												chosen_vertices.add(A  )

												# Remove all arcs adjacent to argmax
												for elem in queue:
																		# if v haven't adjacent node, skip
																		if elem[0] == 0:
																								continue
																		# if argmax is reachable from elem
																		# remove argmax from elem's adjacent list and update his rank
																		if argmax in elem[1][1]:
																								UpperCAmelCase         =			elem[1][1].index(A  )
																								del elem[1][1][index]
																								elem[0] += 1
        # re-order the queue
												heapq.heapify(A  )
						return chosen_vertices


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	_lowercase    :    Optional[int]			     =  {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
	print(F"""Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}""")





