Meertens number explained

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

n be a natural number. We define the Meertens function for base b > 1 F _{b} : N → N to be the following:F _{b} (n) = \prod ^{k -1} _{i =0} p ^{d _{i}} _{k - i -1} . 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} = (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 ^{k(n)} _{b} = 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 ^{i} _{b} (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
	10, 110, 1010		n < 2 ^{96}
	101	11 → 20 → 11	n < 3 ^{60}
	3020	2 → 10 → 2	n < 4 ^{48}
	11, 3032000, 21302000		n < 5 ^{41}
	130	12 → 30 → 12	n < 6 ^{37}
7	202		n < 7 ^{34}
	330		n < 8 ^{32}
	7810000		n < 9 ^{30}
	81312000		n < 10 ^{29}
11	\varnothing		n < 11 ^{44}
	\varnothing		n < 12 ^{40}
13	\varnothing		n < 13 ^{39}
14	13310		n < 14 ^{25}
15	\varnothing		n < 15 ^{37}
	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

References

1. . 1998 . Meertens number . . 8 . 1 . 83–88 . 10.1017/S0956796897002931. 2939112 .