Meertens number

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)=\prod _{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.