code: def groups_count(n):
      r"""Number of groups of order `n`.
      In [1]_, ``gnu(n)`` is given, so we follow this notation here as well.
  
      Parameters
      ==========
  
      n : Integer
          ``n`` is a positive integer
  
      Returns
      =======
  
      int : ``gnu(n)``
  
      Raises
      ======
  
      ValueError
          Number of groups of order ``n`` is unknown or not implemented.
          For example, gnu(`2^{11}`) is not yet known.
          On the other hand, gnu(12) is known to be 5,
          but this has not yet been implemented in this function.
  
      Examples
      ========
  
      >>> from sympy.combinatorics.group_numbers import groups_count
      >>> groups_count(3) # There is only one cyclic group of order 3
      1
      >>> # There are two groups of order 10: the cyclic group and the dihedral group
      >>> groups_count(10)
      2
  
      See Also
      ========
  
      is_cyclic_number
          `n` is cyclic iff gnu(n) = 1
  
      References
      ==========
  
      .. [1] John H. Conway, Heiko Dietrich and E.A. O'Brien,
             Counting groups: gnus, moas and other exotica
             The Mathematical Intelligencer 30, 6-15 (2008)
             https://doi.org/10.1007/BF02985731
      .. [2] https://oeis.org/A000001
  
      """
      n = as_int(n)
      if n <= 0:
          raise ValueError("n must be a positive integer, not %i" % n)
      factors = factorint(n)
      if len(factors) == 1:
          (p, e) = list(factors.items())[0]
          if p == 2:
              A000679 = [
                  1,
                  1,
                  2,
                  5,
                  14,
                  51,
                  267,
                  2328,
                  56092,
                  10494213,
                  49487367289,
              ]
              if e < len(A000679):
                  return A000679[e]
          if p == 3:
              A090091 = [
                  1,
                  1,
                  2,
                  5,
                  15,
                  67,
                  504,
                  9310,
                  1396077,
                  5937876645,
              ]
              if e < len(A090091):
                  return A090091[e]
          if e <= 2:  # gnu(p) = 1, gnu(p**2) = 2
              return e
          if e == 3:  # gnu(p**3) = 5
              return 5
          if e == 4:  # if p is an odd prime, gnu(p**4) = 15
              return 15
          if (
              e == 5
          ):  # if p >= 5, gnu(p**5) is expressed by the following equation
              return 61 + 2 * p + 2 * gcd(p - 1, 3) + gcd(p - 1, 4)
          if (
              e == 6
          ):  # if p >= 6, gnu(p**6) is expressed by the following equation
              return (
                  3 * p**2
                  + 39 * p
                  + 344
                  + 24 * gcd(p - 1, 3)
                  + 11 * gcd(p - 1, 4)
                  + 2 * gcd(p - 1, 5)
              )
          if (
              e == 7
          ):  # if p >= 7, gnu(p**7) is expressed by the following equation
              if p == 5:
                  return 34297
              return (
                  3 * p**5
                  + 12 * p**4
                  + 44 * p**3
                  + 170 * p**2
                  + 707 * p
                  + 2455
                  + (4 * p**2 + 44 * p + 291) * gcd(p - 1, 3)
                  + (p**2 + 19 * p + 135) * gcd(p - 1, 4)
                  + (3 * p + 31) * gcd(p - 1, 5)
                  + 4 * gcd(p - 1, 7)
                  + 5 * gcd(p - 1, 8)
                  + gcd(p - 1, 9)
              )
      if any(e > 1 for e in factors.values()):  # n is not squarefree
          # some known values for small n that have more than 1 factor and are not square free (https://oeis.org/A000001)
          small = {
              12: 5,
              18: 5,
              20: 5,
              24: 15,
              28: 4,
              36: 14,
              40: 14,
              44: 4,
              45: 2,
              48: 52,
              50: 5,
              52: 5,
              54: 15,
              56: 13,
              60: 13,
              63: 4,
              68: 5,
              72: 50,
              75: 3,
              76: 4,
              80: 52,
              84: 15,
              88: 12,
              90: 10,
              92: 4,
          }
          if n in small:
              return small[n]
          raise ValueError(
              "Number of groups of order n is unknown or not implemented"
          )
      if len(factors) == 2:  # n is squarefree semiprime
          p, q = list(factors.keys())
          if p > q:
              p, q = q, p
          return 2 if q % p == 1 else 1
      return _holder_formula(set(factors.keys()))
  
file_path: sympy/sympy/combinatorics/group_numbers.py
imports: from itertools import chain, combinations
  from sympy.external.gmpy import gcd
  from sympy.ntheory.factor_ import factorint
  from sympy.utilities.misc import as_int
label_desc: Use image node_id 6 for calling a global function with example usage: groups_count(n) and returns: _holder_formula, unknown, e, int, int, unknown, unknown, unknown, small, A000679, A090091, int