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 -th 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 ''Fb'' for specific ''b''
All numbers are in base.| Meertens numbers | Cycles | Comments | |
| 2 | 10, 110, 1010 | ||
| 3 | 101 | 11 → 20 → 11 | |
| 4 | 3020 | 2 → 10 → 2 | |
| 5 | 11, 3032000, 21302000 | ||
| 6 | 130 | 12 → 30 → 12 | |
| 7 | 202 | ||
| 8 | 330 | ||
| 9 | 7810000 | ||
| 10 | 81312000 | ||
| 11 | |||
| 12 | |||
| 13 | |||
| 14 | 13310 | ||
| 15 | |||
| 16 | 12 | 2 → 4 → 10 → 2 |