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

Mon Mar 2 18:18:35 CET 2020

```

-----Original Message-----
From: Frank Adams-watters <franktaw at netscape.net>
To: njasloane <njasloane at gmail.com>
Sent: Mon, Mar 2, 2020 11:17 am
Subject: Re: [seqfan] Canonical bijection from positive integers to positive rationals.

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 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.  And would not have much effect.
For me, without doubt,  the best map is the classic map based on the Stern's diatomic series (or Stern-Brocot sequence,  from 1858: A002487.  Look at all the references there.
Best regardsNeil
Neil J. A. Sloane, President, OEIS Foundation.11 South Adelaide Avenue, Highland Park, NJ 08904, USA.Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.Phone: 732 828 6098; home page: http://NeilSloane.comEmail: njasloane at gmail.com

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.