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.[1]

## Definition

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:

${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}p_{k-i-1}^{d_{i)).}$

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 ${\displaystyle i}$-prime number, and

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1))}-n{\bmod {b))^{i)){b^{i))))$

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

${\displaystyle 3020=2^{3}3^{0}5^{2}7^{0))$.

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.

## Meertens numbers and cycles of ${\displaystyle F_{b))$ for specific ${\displaystyle b}$

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

${\displaystyle b}$ Meertens numbers Cycles Comments
2 10, 110, 1010 ${\displaystyle n<2^{96))$[2]
3 101 11 → 20 → 11 ${\displaystyle n<3^{60))$[2]
4 3020 2 → 10 → 2 ${\displaystyle n<4^{48))$[2]
5 11, 3032000, 21302000 ${\displaystyle n<5^{41))$[2]
6 130 12 → 30 → 12 ${\displaystyle n<6^{37))$[2]
7 202 ${\displaystyle n<7^{34))$[2]
8 330 ${\displaystyle n<8^{32))$[2]
9 7810000 ${\displaystyle n<9^{30))$[2]
10 81312000 ${\displaystyle n<10^{29))$[2]
11 ${\displaystyle \varnothing }$ ${\displaystyle n<11^{44))$[2]
12 ${\displaystyle \varnothing }$ ${\displaystyle n<12^{40))$[2]
13 ${\displaystyle \varnothing }$ ${\displaystyle n<13^{39))$[2]
14 13310 ${\displaystyle n<14^{25))$[2]
15 ${\displaystyle \varnothing }$ ${\displaystyle n<15^{37))$[2]
16 12 2 → 4 → 10 → 2 ${\displaystyle n<16^{24))$[2]