Meertens number

In number theory and mathematical logic, a Meertens number in a given number base formula_1 is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam.
Definition.
Let formula_2 be a natural number. We define the Meertens function for base formula_3 formula_4 to be the following:
where formula_6 is the number of digits in the number in base formula_1, formula_8 is the formula_9-prime number, and 
is the value of each digit of the number. A natural number formula_2 is a Meertens number if it is a fixed point for formula_12, which occurs if formula_13. This corresponds to a Gödel encoding.
For example, the number 3020 in base formula_14 is a Meertens number, because 
A natural number formula_2 is a sociable Meertens number if it is a periodic point for formula_12, where formula_18 for a positive integer formula_19, and forms a cycle of period formula_19. A Meertens number is a sociable Meertens number with formula_21, and a amicable Meertens number is a sociable Meertens number with formula_22.
The number of iterations formula_9 needed for formula_24 to reach a fixed point is the Meertens function's persistence of formula_2, and undefined if it never reaches a fixed point.
Meertens numbers and cycles of formula_26 for specific formula_1.
All numbers are in base formula_1.