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 for specific All numbers are in base . 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 References External links Arithmetic dynamics Base-dependent integer sequences