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 ${\displaystyle b>1}$ ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

${\displaystyle F_{b}(n)=\prod _{i=0}^{k-1}p_{k-i-1}^{d_{i}}.}$

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 ${\displaystyle i}$-th prime number (starting at 0), 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 Fb for specific b

All numbers are in base ${\displaystyle b}$.

${\displaystyle b}$	Meertens numbers	Cycles	Comments
2	10, 110, 1010		${\displaystyle n<2^{96}}$[2]
3	101	11 → 20 → 11	${\displaystyle n<3^{60}}$[2]
4	3020	2 → 10 → 2	${\displaystyle n<4^{48}}$[2]
5	11, 3032000, 21302000		${\displaystyle n<5^{41}}$[2]
6	130	12 → 30 → 12	${\displaystyle n<6^{37}}$[2]
7	202		${\displaystyle n<7^{34}}$[2]
8	330		${\displaystyle n<8^{32}}$[2]
9	7810000		${\displaystyle n<9^{30}}$[2]
10	81312000		${\displaystyle n<10^{29}}$[2]
11	${\displaystyle \varnothing }$		${\displaystyle n<11^{44}}$[2]
12	${\displaystyle \varnothing }$		${\displaystyle n<12^{40}}$[2]
13	${\displaystyle \varnothing }$		${\displaystyle n<13^{39}}$[2]
14	13310		${\displaystyle n<14^{25}}$[2]
15	${\displaystyle \varnothing }$		${\displaystyle n<15^{37}}$[2]
16	12	2 → 4 → 10 → 2	${\displaystyle n<16^{24}}$[2]

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