Meertens number


In number theory and mathematical logic, a Meertens number in a given number base 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 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 F_b for specific b

All numbers are in base.
Meertens numbersCyclesComments
210, 110, 1010
310111 → 20 → 11
430202 → 10 → 2
511, 3032000, 21301100
613012 → 30 → 12
7202
8330
97810000
1081312000
11
12
13
1413310
15
16122 → 4 → 10 → 2