Meertens number

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In number theory and mathematical logic, a Meertens number in a given number base 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[edit]

Let be a natural number. We define the Meertens function for base to be the following:

where is the number of digits in the number in base , is the -prime number, and

is the value of each digit of the number. A natural number is a Meertens number if it is a fixed point for , which occurs if . This corresponds to a Gödel encoding.

For example, the number 3020 in base is a Meertens number, because

.

A natural number is a sociable Meertens number if it is a periodic point for , where for a positive integer , and forms a cycle of period . A Meertens number is a sociable Meertens number with , and a amicable Meertens number is a sociable Meertens number with .

The number of iterations needed for to reach a fixed point is the Meertens function's persistence of , and undefined if it never reaches a fixed point.

Meertens numbers and cycles of for specific [edit]

All numbers are in base .

Meertens numbers Cycles Comments
2 10, 110, 1010 [2]
3 101 11 → 20 → 11 [2]
4 3020 2 → 10 → 2 [2]
5 11, 3032000, 21301100 [2]
6 130 12 → 30 → 12 [2]
7 202 [2]
8 330 [2]
9 7810000 [2]
10 81312000 [2]
11 [2]
12 [2]
13 [2]
14 13310 [2]
15 [2]
16 12 2 → 4 → 10 → 2 [2]

See also[edit]

References[edit]

  1. ^ Richard S. Bird (1998). "Meertens number". Journal of Functional Programming. 8 (1): 83–88. doi:10.1017/S0956796897002931.
  2. ^ a b c d e f g h i j k l m n o (sequence A246532 in the OEIS)

External links[edit]