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.

be a natural number.

We define the Meertens function for base

A natural number

is a Meertens number if it is a fixed point for

This corresponds to a Gödel encoding.

For example, the number 3020 in base

is a sociable Meertens number if it is a periodic point for

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

to reach a fixed point is the Meertens function's persistence of

, and undefined if it never reaches a fixed point.

All numbers are in base