In number theory and mathematical logic, a Meertens number in a given number base $b$ 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.

## Definition

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

$F_{b}(n)=\sum _{i=0}^{k-1}p_{k-i-1}^{d_{i)).$ where $k=\lfloor \log _{b}{n}\rfloor +1$ is the number of digits in the number in base $b$ , $p_{i)$ is the $i$ -prime number, and

$d_{i}={\frac {n{\bmod {b^{i+1))}-n{\bmod {b))^{i)){b^{i)))$ is the value of each digit of the number. A natural number $n$ is a Meertens number if it is a fixed point for $F_{b)$ , which occurs if $F_{b}(n)=n$ . This corresponds to a Gödel encoding.

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

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

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

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

## Meertens numbers and cycles of $F_{b)$ for specific $b$ All numbers are in base $b$ .

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