






import math


def      lowerCamelCase__	(   lowercase ):

  """simple docstring"""






  if n <= 0:
    raise ValueError("Only positive integers have prime factors" )
  SCREAMING_SNAKE_CASE				:					int		  =			[]
  while n % 2 == 0:
    pf.append(2 )
    SCREAMING_SNAKE_CASE				:					List[str]		  =			int(n / 2 )
  for i in range(3  ,   int(math.sqrt(lowercase ) ) + 1  ,   2 ):
    while n % i == 0:
      pf.append(lowercase )
      SCREAMING_SNAKE_CASE				:					int		  =			int(n / i )
  if n > 2:
    pf.append(lowercase )
  return pf


def      lowerCamelCase__	(   lowercase ):

  """simple docstring"""






  if n <= 0:
    raise ValueError("Only positive numbers are accepted" )
  SCREAMING_SNAKE_CASE				:					Dict		  =			1
  SCREAMING_SNAKE_CASE				:					Dict		  =			1
  while n % 2 == 0:
    temp += 1
    SCREAMING_SNAKE_CASE				:					int		  =			int(n / 2 )
  div *= temp
  for i in range(3  ,   int(math.sqrt(lowercase ) ) + 1  ,   2 ):
    SCREAMING_SNAKE_CASE				:					int		  =			1
    while n % i == 0:
      temp += 1
      SCREAMING_SNAKE_CASE				:					Optional[int]		  =			int(n / i )
    div *= temp
  if n > 1:
    div *= 2
  return div


def      lowerCamelCase__	(   lowercase ):

  """simple docstring"""






  if n <= 0:
    raise ValueError("Only positive numbers are accepted" )
  SCREAMING_SNAKE_CASE				:					Optional[Any]		  =			1
  SCREAMING_SNAKE_CASE				:					int		  =			1
  while n % 2 == 0:
    temp += 1
    SCREAMING_SNAKE_CASE				:					Dict		  =			int(n / 2 )
  if temp > 1:
    s *= (2**temp - 1) / (2 - 1)
  for i in range(3  ,   int(math.sqrt(lowercase ) ) + 1  ,   2 ):
    SCREAMING_SNAKE_CASE				:					Any		  =			1
    while n % i == 0:
      temp += 1
      SCREAMING_SNAKE_CASE				:					Optional[int]		  =			int(n / i )
    if temp > 1:
      s *= (i**temp - 1) / (i - 1)
  return int(lowercase )


def      lowerCamelCase__	(   lowercase ):

  """simple docstring"""






  SCREAMING_SNAKE_CASE				:					str		  =			n
  for x in set(prime_factors(lowercase ) ):
    s *= (x - 1) / x
  return int(lowercase )


if __name__ == "__main__":
   import doctest

   doctest.testmod()








'''simple docstring'''
def 						A_(			A		:						list[list[int]]):
							if len(A) == 0 or not all(len(A) == 2 for pair in coordinates):
														raise ValueError('The program cannot work out a fitting polynomial.')

							if len({tuple(A) for pair in coordinates}) != len(A):
														raise ValueError('The program cannot work out a fitting polynomial.')

							UpperCamelCase					       =			{x for x, _ in coordinates}
							if len(A) == 1:
														return f'''x={coordinates[0][0]}'''

							if len(A) != len(A):
														raise ValueError('The program cannot work out a fitting polynomial.')

							UpperCamelCase					       =			len(A)

							UpperCamelCase					       =			0
							UpperCamelCase					       =			[]
							# put the x and x to the power values in a matrix
							while count_of_line < x:
														UpperCamelCase					       =			0
														UpperCamelCase					       =			coordinates[count_of_line][0]
														UpperCamelCase					       =			[]
														while count_in_line < x:
																					count_line.append(a ** (x - (count_in_line + 1)))
																					count_in_line += 1
														matrix.append(A)
														count_of_line += 1

							UpperCamelCase					       =			0
							# put the y values into a vector
							UpperCamelCase					       =			[]
							while count_of_line < x:
														vector.append(coordinates[count_of_line][1])
														count_of_line += 1

							UpperCamelCase					       =			0

							while count < x:
														UpperCamelCase					       =			0
														while zahlen < x:
																					if count == zahlen:
																												zahlen += 1
																					if zahlen == x:
																												break
																					UpperCamelCase					       =			matrix[zahlen][count] / matrix[count][count]
																					for counting_columns, item in enumerate(matrix[count]):
																												# manipulating all the values in the matrix
																												matrix[zahlen][counting_columns] -= item * bruch
																					# manipulating the values in the vector
																					vector[zahlen] -= vector[count] * bruch
																					zahlen += 1
														count += 1

							UpperCamelCase					       =			0
							# make solutions
							UpperCamelCase					       =			[]
							while count < x:
														solution.append(str(vector[count] / matrix[count][count]))
														count += 1

							UpperCamelCase					       =			0
							UpperCamelCase					       =			'f(x)='

							while count < x:
														UpperCamelCase					       =			solution[count].split('E')
														if len(A) > 1:
																					UpperCamelCase					       =			f'''{remove_e[0]}*10^{remove_e[1]}'''
														solved += f'''x^{x - (count + 1)}*{solution[count]}'''
														if count + 1 != x:
																					solved += "+"
														count += 1

							return solved


if __name__ == "__main__":
						print(points_to_polynomial([]))
						print(points_to_polynomial([[]]))
						print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
						print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
						print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
						print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
						print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
						print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))




