Meertens number

Summary

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 Fb for specific b 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, 21302000  [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. S2CID 2939112.
  2. ^ a b c d e f g h i j k l m n o (sequence A246532 in the OEIS)

External links edit

  • 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.)