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 (starting at 0), 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
See also Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number
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