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:
 * $$F_{b}(n) = \sum_{i=0}^{k - 1} p_{k - i - 1}^{d_i}. $$

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 Fb for specific b
All numbers are in base $$b$$.