# [seqfan] Re: A new (or old), encompassing definition of Integer Sequences.

Maximilian Hasler maximilian.hasler at gmail.com
Mon Feb 11 16:29:34 CET 2013

```On Sun, Feb 10, 2013 at 4:39 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> An integer sequence is a function Z -> Z (usually partial),
> whose domain (*) is defined by an integer sequence.

I don't like that too much...
That's much "noise" / big machinery, mainly to deal with a tiny offset
(I know you also think of "holes" in the domain, but see below...)

In addition, it is ill defined (through an infinite recursion)... ;-) !

> (* in the first sense given in
> http://en.wikipedia.org/wiki/Partial_function : "Most mathematicians,
> including recursion theorists, use the term "domain of f" for the set
> of all values x such that f(x) is defined.")

here I totally agree, not much doubt...

> How about that?
>
> Why so general? Then we can include also David Wilson's idea from
> 1999: http://list.seqfan.eu/pipermail/seqfan/2012-March/009214.html
>
> Actually, now that I re-read it, it seems that his "index sequences"
> is just the same as what I mean by the domain sequence here.

well, yes and no...
If you define:
"a subset of the integers [bounded from below] is a strictly
increasing sequence"
then one could argue that it might be the same,
but the philosophy is a bit different ; requiring a "sequence" instead
of a domain implies a re-indexing of the original sequence..

In that sense, it is not "so general" but rather restrictive:
instead of allowing sequences with holes, you impose in some sense
that only integer sequences defined on IN = {1,2,3...} are allowed,
and if this is not the case a priori,
then the "faulty" sequence must be redefined to be the composition of
itself with the sequence listing its domain by indexing it with IN
...

> Note that the domains of the most infinite sequences in OEIS
> have currently been defined with either A001477 or A000027.
> (or with their characteristic functions on Z, if you so wish.)

yes and no (again);
there are many many sequences of the form
"a(n) = [some property] of the n-th prime"
or similar.
In your philosophy, should this be "forbidden" in favour of defining
that sequence
as a sequence defined "on the primes only" ?
i.e., "a(p) = [some property] of the prime p"
and be re-indexed by the sequence A40 (the primes) ?

Maximilian
```