code: def test_smoothness_and_smoothness_p():
      assert smoothness(1) == (1, 1)
      assert smoothness(2**4 * 3**2) == (3, 16)
  
      assert smoothness_p(10431, m=1) == (
          1,
          [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))],
      )
      assert smoothness_p(10431) == (
          -1,
          [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))],
      )
      assert smoothness_p(10431, power=1) == (
          -1,
          [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))],
      )
      assert (
          smoothness_p(21477639576571, visual=1)
          == "p**i=4410317**1 has p-1 B=1787, B-pow=1787\n"
          + "p**i=4869863**1 has p-1 B=2434931, B-pow=2434931"
      )
  
imports: from sympy.concrete.summations import summation
  from sympy.core.containers import Dict
  from sympy.core.mul import Mul
  from sympy.core.power import Pow
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.core.numbers import Integer, Rational
  from sympy.external.gmpy import gcd
  from sympy.ntheory import totient, factorint, primefactors, divisors, nextprime, primerange, pollard_rho, perfect_power, multiplicity, multiplicity_in_factorial, divisor_count, primorial, pollard_pm1, divisor_sigma, factorrat, reduced_totient
  from sympy.ntheory.factor_ import smoothness, smoothness_p, proper_divisors, antidivisors, antidivisor_count, core, udivisors, udivisor_sigma, udivisor_count, proper_divisor_count, primenu, primeomega, mersenne_prime_exponent, is_perfect, is_abundant, is_deficient, is_amicable, dra, drm, _perfect_power
  from sympy.testing.pytest import raises, slow
  from sympy.utilities.iterables import capture
label_desc: Use image node_id 23 for calling a global function with example usage: test_smoothness_and_smoothness_p() without return types