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.

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	$n<16^{24}$

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

External links

• OEIS sequence A189398 (a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k))
• OEIS sequence A246532 (Smallest Meertens number in base n, or -1 if none exists.)