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 $b > 1$ $F_ : \mathbb \rightarrow \mathbb$ to be the following:
:$F_(n) = \sum_^ p_^.$
where $k = \lfloor \log_ \rfloor + 1$ is the number of digits in the number in base $b$, $p_i$ is the $i$-prime number, and
:$d_i = \frac$
is the value of each digit of the number. A natural number $n$ is a Meertens number if it is a fixed point for $F_$, which occurs if $F_(n) = n$. This corresponds to a Gödel encoding.

For example, the number 3020 in base $b = 4$ is a Meertens number, because
:$3020 = 2^3^5^7^$.

A natural number $n$ is a sociable Meertens number if it is a periodic point for $F_$, where $F_^k(n) = n$ 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 $F_^(n)$ 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 $b$.

See also

* Arithmetic dynamics
* Dudeney number
* Factorion
* Happy number
* Kaprekar's constant
* Kaprekar number
* Narcissistic number
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Sum-product number

References

External links

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{{Classes of natural numbers

Arithmetic dynamics
Base-dependent integer sequences