# [seqfan] Re: Fwd: Canonical bijection from positive integers to positive rationals.

Peter Munn techsubs at pearceneptune.co.uk
Wed Mar 4 16:28:24 CET 2020

```On Mon, March 2, 2020 5:18 pm, Frank Adams-watters via SeqFan wrote:
> From: Frank Adams-watters <franktaw at netscape.net>
> To: njasloane <njasloane at gmail.com>
> Sent: Mon, Mar 2, 2020 11:17 am
>
> Let me back off a minute here. I am planning on submitting a number of
> sequences encapsulating functions on positive rationals. Do we have a
> standard way to do this? If not, what should the standard be?
>
> I would think that one needs to choose a bijection s: Z+ -> Q+, and then
> if the function is f() the sequence is a(n) = f(s(n)). It was my
> understanding that the "canonical" bijection was precisely the s() to use
> for this. If not, then I don't understand what "canonical" means in a
> sequence.

I can see that it would be useful for there not to be a lot of different
mappings, s: Z+ -> Q+ used in OEIS for functions on rationals. For one
thing, it would make it difficult to find them by searching for the
sequence terms.

So, I decided to see if I could find any of the following sequences in
OEIS by looking for terms I had calculated:
a(n) = s^-1(s(n) * 2)
a(n) = s^-1(s(n) / 2)
T(n,k) = s^-1(s(n) + s(k))
T(n,k) = s^-1(s(n) * s(k)).

I did this for s corresponding to A002487, A020652/A020653,
A038568/A038569, A071974/A071975 and A229994/A077610.

I drew a blank. (I gave up at that point, so I didn't try A295515 or any
others. I had looked for both square and triangular forms of tables.)

A different search technique could prove more fruitful, but unless anyone
knows better, it may be you are heading into virgin territory, Franklin.

FWIW, the mapping that seemed best at keeping the early terms small (in
the above sequences) was A229994/A077610.

Best Regards,

Peter

> I am familiar with Stern-Brocot, and I agree that it is very beautiful. I
> didn't include it in the original message because it doesn't claim to be
> canonical. If we wind up using it for this purpose, I will have no
> objection.
>
>
> -----Original Message-----
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Cc: Frank Adams-watters <franktaw at netscape.net>
> Sent: Mon, Mar 2, 2020 10:28 am
> Subject: Re: [seqfan] Canonical bijection from positive integers to
> positive rationals.
>
> I don't think we need to choose an official OEIS map, that would be
> presumptuous.Â  And would not have much effect.
> For me, without doubt, Â the best map is the classic map based on the
> Stern's diatomic series (or Stern-Brocot sequence, Â from 1858: A002487.Â
> Look at all the references there.
> Best regardsNeilÂ
> Neil J. A. Sloane, President, OEIS Foundation.
>
> On Mon, Mar 2, 2020 at 11:14 AM Frank Adams-watters via SeqFan
> <seqfan at list.seqfan.eu> wrote:
> There are two pairs of sequences claiming to be the canonical bijection
> from positive integers to positive rationals:
>
> A020652/A020653, and
> A038568/A038569.
>
> Which do we want to actually be our canonical sequence?
>
> I favor A020652/A020653. It seems cleaner to me.
>
> Compare A038566/A038567, which is the basis for both of these.

```