Riffs & Rotes

[Jon Awbrey]

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R&R.  Note 16

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SeqFans,

Here's a more compact description of the primal code sequences:

Taking the somewhat PID-headed point of view that the purpose
of the positive integers, whoever created them, is to codify
finite partial functions from themselves into themselves,
let's call the thus-denoted object the "primal function"
of the corresponding positive integer code.

Given a set of functions, it is natural to compose them.
Let "o" denote ordinary functional composition, and let
"n o m" denote the composition of the primal functions
of n and m.  Composing the primal functions of positive
integers yields the primal function composition Tables
that are given with the following couple of sequences:

   A106177.  http://www.research.att.com/projects/OEIS?Anum=A106177
   A106178.  http://www.research.att.com/projects/OEIS?Anum=A106178

The k^th compositional power of the primal function of n is written as
(n o)^k and defined as (n o)^k = n o ... o n, with k occurrences of n.
The initial values of (n o)^k are given in a Table with this sequence:

   A108371.  http://www.research.att.com/projects/OEIS?Anum=A108371
    
The primal code characteristic of a positive integer n is
the least positive integer k, if any, such that (n o)^k = 1,
otherwise set equal to 0 if no such k exists.  The sequence
of primal code characterictics of positive integers is here:

   A108352.  http://www.research.att.com/projects/OEIS?Anum=A108352

The primal functions of the first 1200 positive integers,
their primal code characteristics, and other useful data
are arrayed in (the first draft of) a Table at this site:

   R&R 15.  http://stderr.org/pipermail/inquiry/2005-June/002799.html

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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