Meertens number

In number theory and mathematical logic, a Meertens number in a given number base [math]\displaystyle{ b }[/math] 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 [math]\displaystyle{ n }[/math] be a natural number. We define the Meertens function for base [math]\displaystyle{ b \gt 1 }[/math] [math]\displaystyle{ F_{b} : \mathbb{N} \rightarrow \mathbb{N} }[/math] to be the following:

[math]\displaystyle{ F_{b}(n) = \sum_{i=0}^{k - 1} p_{k - i - 1}^{d_i}. }[/math]

where [math]\displaystyle{ k = \lfloor \log_{b}{n} \rfloor + 1 }[/math] is the number of digits in the number in base [math]\displaystyle{ b }[/math], [math]\displaystyle{ p_i }[/math] is the [math]\displaystyle{ i }[/math]-prime number, and

[math]\displaystyle{ d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} }[/math]

is the value of each digit of the number. A natural number [math]\displaystyle{ n }[/math] is a Meertens number if it is a fixed point for [math]\displaystyle{ F_{b} }[/math], which occurs if [math]\displaystyle{ F_{b}(n) = n }[/math]. This corresponds to a Gödel encoding.

For example, the number 3020 in base [math]\displaystyle{ b = 4 }[/math] is a Meertens number, because

[math]\displaystyle{ 3020 = 2^{3}3^{0}5^{2}7^{0} }[/math].

A natural number [math]\displaystyle{ n }[/math] is a sociable Meertens number if it is a periodic point for [math]\displaystyle{ F_{b} }[/math], where [math]\displaystyle{ F_{b}^k(n) = n }[/math] for a positive integer [math]\displaystyle{ k }[/math], and forms a cycle of period [math]\displaystyle{ k }[/math]. A Meertens number is a sociable Meertens number with [math]\displaystyle{ k = 1 }[/math], and a amicable Meertens number is a sociable Meertens number with [math]\displaystyle{ k = 2 }[/math].

The number of iterations [math]\displaystyle{ i }[/math] needed for [math]\displaystyle{ F_{b}^{i}(n) }[/math] to reach a fixed point is the Meertens function's persistence of [math]\displaystyle{ n }[/math], and undefined if it never reaches a fixed point.

Meertens numbers and cycles of F_b for specific b

All numbers are in base [math]\displaystyle{ b }[/math].

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

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.)