code: def test_divisor_sigma():
      assert [divisor_sigma(k) for k in range(1, 12)] == [
          1,
          3,
          4,
          7,
          6,
          12,
          8,
          15,
          13,
          18,
          12,
      ]
      assert [divisor_sigma(k, 2) for k in range(1, 12)] == [
          1,
          5,
          10,
          21,
          26,
          50,
          50,
          85,
          91,
          130,
          122,
      ]
      assert divisor_sigma(23450) == 50592
      assert divisor_sigma(23450, 0) == 24
      assert divisor_sigma(23450, 1) == 50592
      assert divisor_sigma(23450, 2) == 730747500
      assert divisor_sigma(23450, 3) == 14666785333344
  
      a = Symbol("a", prime=True)
      b = Symbol("b", prime=True)
      j = Symbol("j", integer=True, positive=True)
      k = Symbol("k", integer=True, positive=True)
      assert divisor_sigma(a**j * b**k) == (a ** (j + 1) - 1) * (
          b ** (k + 1) - 1
      ) / ((a - 1) * (b - 1))
      assert divisor_sigma(a**j * b**k, 2) == (
          a ** (2 * j + 2) - 1
      ) * (b ** (2 * k + 2) - 1) / ((a**2 - 1) * (b**2 - 1))
      assert divisor_sigma(a**j * b**k, 0) == (j + 1) * (k + 1)
  
      m = Symbol("m", integer=True)
      k = Symbol("k", integer=True)
      assert divisor_sigma(m)
      assert divisor_sigma(m, k)
      assert divisor_sigma(m).subs(m, 3**10) == 88573
      assert (
          divisor_sigma(m, k).subs([(m, 3**10), (k, 3)])
          == 213810021790597
      )
      assert summation(divisor_sigma(m), (m, 1, 11)) == 99
  
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 14 for calling a global function with example usage: test_divisor_sigma() without return types