




import math
def 		a_       (   __lowerCAmelCase							):
  if n <= 0:
    raise ValueError('''Only positive integers have prime factors'''							)
  lowerCAmelCase__								=			[]
  while n % 2 == 0:
    pf.append(2							)
    lowerCAmelCase__								=			int(n / 2							)
  for i in range(3      ,   int(math.sqrt(__lowerCAmelCase							)							) + 1      ,   2							):
    while n % i == 0:
      pf.append(__lowerCAmelCase							)
      lowerCAmelCase__								=			int(n / i							)
  if n > 2:
    pf.append(__lowerCAmelCase							)
  return pf
def 		a_       (   __lowerCAmelCase							):
  if n <= 0:
    raise ValueError('''Only positive numbers are accepted'''							)
  lowerCAmelCase__								=			1
  lowerCAmelCase__								=			1
  while n % 2 == 0:
    temp += 1
    lowerCAmelCase__								=			int(n / 2							)
  div *= temp
  for i in range(3      ,   int(math.sqrt(__lowerCAmelCase							)							) + 1      ,   2							):
    lowerCAmelCase__								=			1
    while n % i == 0:
      temp += 1
      lowerCAmelCase__								=			int(n / i							)
    div *= temp
  if n > 1:
    div *= 2
  return div
def 		a_       (   __lowerCAmelCase							):
  if n <= 0:
    raise ValueError('''Only positive numbers are accepted'''							)
  lowerCAmelCase__								=			1
  lowerCAmelCase__								=			1
  while n % 2 == 0:
    temp += 1
    lowerCAmelCase__								=			int(n / 2							)
  if temp > 1:
    s *= (2**temp - 1) / (2 - 1)
  for i in range(3      ,   int(math.sqrt(__lowerCAmelCase							)							) + 1      ,   2							):
    lowerCAmelCase__								=			1
    while n % i == 0:
      temp += 1
      lowerCAmelCase__								=			int(n / i							)
    if temp > 1:
      s *= (i**temp - 1) / (i - 1)
  return int(__lowerCAmelCase							)
def 		a_       (   __lowerCAmelCase							):
  lowerCAmelCase__								=			n
  for x in set(prime_factors(__lowerCAmelCase							)							):
    s *= (x - 1) / x
  return int(__lowerCAmelCase							)


if __name__ == "__main__":
      import doctest

      doctest.testmod()







def 		a_       (   __lowerCAmelCase      ,   __lowerCAmelCase							):
  lowerCAmelCase__								=			0
  for ele in arr:
    total_sum += product_sum(__lowerCAmelCase      ,   depth + 1							) if isinstance(__lowerCAmelCase      ,   __lowerCAmelCase							) else ele
  return total_sum * depth
def 		a_       (   __lowerCAmelCase							):
  return product_sum(__lowerCAmelCase      ,   1							)


if __name__ == "__main__":
      import doctest

      doctest.testmod()

