Factorion

In number theory, a factorion in a given number base formula_1 is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover.
Definition.
Let formula_2 be a natural number. We define the sum of the factorial of the digits of formula_2 for base formula_4 formula_5 to be the following:
where formula_7 is the number of digits in the number in base formula_1, formula_9 is the factorial of formula_2 and 
is the value of each digit of the number. A natural number formula_2 is a formula_1-factorion if it is a fixed point for formula_14, which occurs if formula_15. formula_16 and formula_17 are fixed points for all formula_1, and thus are trivial factorions for all formula_1, and all other factorions are nontrivial factorions. 
For example, the number 145 in base formula_20 is a factorion because formula_21. 
For formula_22, the sum of the factorial of the digits is simply the number of digits formula_23 in the base 2 representation. 
A natural number formula_2 is a sociable factorion if it is a periodic point for formula_14, where formula_26 for a positive integer formula_23, and forms a cycle of period formula_23. A factorion is a sociable factorion with formula_29, and a amicable factorion is a sociable factorion with formula_30.
All natural numbers formula_2 are preperiodic points for formula_14, regardless of the base. This is because all natural numbers of base formula_1 with formula_23 digits satisfy formula_35. However, when formula_36, then formula_37 for formula_38, so any formula_2 will satisfy formula_40 until formula_41. There are a finite number of natural numbers less than formula_42, so the number is guaranteed to reach a periodic point or a fixed point less than formula_43, making it a preperiodic point. For formula_22, the number of digits formula_45 for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base formula_1. 
The number of iterations formula_47 needed for formula_48 to reach a fixed point is the formula_14 function's persistence of formula_2, and undefined if it never reaches a fixed point.
Factorions for formula_14.
b = (k - 1)!
Let formula_23 be a positive integer and the number base formula_53. Then:
b = k! - k + 1.
Let formula_23 be a positive integer and the number base formula_61. Then:
Table of factorions and cycles of formula_14.
All numbers are represented in base formula_1. 
Programming example.
The example below implements the sum of the factorial of the digits described in the definition above to search for factorions and cycles in Python.
def factorial(x: int) -> int:
 total = 1
 for i in range(0, x):
 total = total * (i + 1)
 return total
def sfd(x: int, b: int) -> int:
 """Sum of the factorial of the digits."""
 total = 0
 while x > 0:
 total = total + factorial(x % b)
 x = x // b
 return total
def sfd_cycle(x: int, b: int) -> List[int]:
 seen = []
 while x not in seen:
 seen.append(x)
 x = sfd(x, b)
 cycle = []
 while x not in cycle:
 cycle.append(x)
 x = sfd(x, b)
 return cycle