# Meertens number

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 $\displaystyle{ b \gt 1 }$ $\displaystyle{ F_{b} : \mathbb{N} \rightarrow \mathbb{N} }$ 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 $\displaystyle{ n \lt 2^{96} }$[2]
3 101 11 → 20 → 11 $\displaystyle{ n \lt 3^{60} }$[2]
4 3020 2 → 10 → 2 $\displaystyle{ n \lt 4^{48} }$[2]
5 11, 3032000, 21302000 $\displaystyle{ n \lt 5^{41} }$[2]
6 130 12 → 30 → 12 $\displaystyle{ n \lt 6^{37} }$[2]
7 202 $\displaystyle{ n \lt 7^{34} }$[2]
8 330 $\displaystyle{ n \lt 8^{32} }$[2]
9 7810000 $\displaystyle{ n \lt 9^{30} }$[2]
10 81312000 $\displaystyle{ n \lt 10^{29} }$[2]
11 $\displaystyle{ \varnothing }$ $\displaystyle{ n \lt 11^{44} }$[2]
12 $\displaystyle{ \varnothing }$ $\displaystyle{ n \lt 12^{40} }$[2]
13 $\displaystyle{ \varnothing }$ $\displaystyle{ n \lt 13^{39} }$[2]
14 13310 $\displaystyle{ n \lt 14^{25} }$[2]
15 $\displaystyle{ \varnothing }$ $\displaystyle{ n \lt 15^{37} }$[2]
16 12 2 → 4 → 10 → 2 $\displaystyle{ n \lt 16^{24} }$[2]