In number theory and mathematical logic, a Meertens number in a given number base ${\displaystyle b}$ 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.

Let ${\displaystyle n}$ be a natural number. We define the Meertens function for base ${\displaystyle b>1}$ ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$ is the number of digits in the number in base ${\displaystyle b}$, ${\displaystyle p_{i}}$ is the ${\displaystyle i}$-th prime number (starting at 0), and

is the value of each digit of the number. A natural number ${\displaystyle n}$ is a Meertens number if it is a fixed point for ${\displaystyle F_{b}}$, which occurs if ${\displaystyle F_{b}(n)=n}$. This corresponds to a Gödel encoding.

For example, the number 3020 in base ${\displaystyle b=4}$ is a Meertens number, because

A natural number ${\displaystyle n}$ is a sociable Meertens number if it is a periodic point for ${\displaystyle F_{b}}$, where ${\displaystyle F_{b}^{k}(n)=n}$ for a positive integer ${\displaystyle k}$, and forms a cycle of period ${\displaystyle k}$. A Meertens number is a sociable Meertens number with ${\displaystyle k=1}$, and a amicable Meertens number is a sociable Meertens number with ${\displaystyle k=2}$.

The number of iterations ${\displaystyle i}$ needed for ${\displaystyle F_{b}^{i}(n)}$ to reach a fixed point is the Meertens function's persistence of ${\displaystyle n}$, and undefined if it never reaches a fixed point.

All numbers are in base ${\displaystyle b}$.