Let be a natural number. We define the Meertens function for base to be the following:
where is the number of digits in the number in base , is the -prime number, and
is the value of each digit of the number. A natural number is a Meertens number if it is a fixed point for , which occurs if . This corresponds to a Gödel encoding.
For example, the number 3020 in base is a Meertens number, because
A natural number is a sociable Meertens number if it is a periodic point for , where for a positive integer , and forms a cycle of period . A Meertens number is a sociable Meertens number with , and a amicable Meertens number is a sociable Meertens number with .
The number of iterations needed for to reach a fixed point is the Meertens function's persistence of , and undefined if it never reaches a fixed point.
Meertens numbers and cycles of for specific