How (not) to deal with conjectures.

[Wouter Meeussen]

Hello seqfanners,

one of the joys of the OEIS, close to it's original purpose, is to be alerted when your sequence occurs in the database under a different definition.
This points to a (probable) equality between two (combinatorial) entities. If true, it might be easy or hard to prove. It might be evident, known, trivial or interesting to those in-the-know.

What to do in such cases?

1/   since it's only a conjecture, and since the editorial consensus is 
(quote)

Programs base on conjecture are disallowed, so this is not ok.
I don't think ok in comments or anywhere else either.
You need to prove the conjecture first. (Until then the lack of a program is not an issue)

(end quote)
then maybe just drop the issue all together.

2/  be obstinate and enter a 'new' sequence with your definition, and add a comment that it might be equal to the existing one you first hit on.
This avoids the obstacle under 1/ above, but makes the (probable) equality harder to find and bloats the database with a duplicate entry.

Suggestions, anyone?

Wouter.

ps. I once added a conjecture to a sequence and later found that D. Knuth completed the sequence with it's (short) proof. That's a rare occurrence.

[Kevin Ryde]

"'Wouter Meeussen' via SeqFan" <seq...@googlegroups.com> writes:
>
> Suggestions, anyone?

Are you alluding to A309097 ?

Looks like partitions permitted there would be any
of 1 or 2 distinct parts, and those of 3 distinct parts
with parts x,y,y+1 with x < y.

Presumably counting multiples gets to a little sum.
Then aim for a sum over 1 variable by something to
do with quotients :).

[Sean A. Irvine]

Hi Wouter,

From my perspective the editorial advice you were given here is correct. It is simply too dangerous for the OEIS to accept conjectural programs because inevitably someone subsequently overlooks the warning and uses it to add more terms, extend a b-file, or mistakenly takes it as fact in the construction of a formula, and so on.

You can add a comment indicating that a sequence is conjectured to be the same as some other interpretation.

Otherwise, yes, you are in situation 2.  But such a sequence is only going to be accepted if the contribution is deemed to be sufficiently interesting or important to warrant such a treatment.

Regards,

Sean.

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[David desJardins]

We should have the empty sequence just so we can add a lot of conjectural notes to it. “Odd perfect numbers”, etc.

[Antti Karttunen]

On Mon, Sep 1, 2025 at 4:32 AM Sean A. Irvine <sai...@gmail.com> wrote:

Hi Wouter,

From my perspective the editorial advice you were given here is correct. It is simply too dangerous for the OEIS to accept conjectural programs because inevitably someone subsequently overlooks the warning and uses it to add more terms, extend a b-file, or mistakenly takes it as fact in the construction of a formula, and so on.

You can add a comment indicating that a sequence is conjectured to be the same as some other interpretation.

On the other hand, I see Wouter's point 1 in those cases where the conjectured new interpretation is something complex (e.g., particular kind of partitions, tilings, polyomino-configurations, etc), that a small program would the best way to express what is exactly meant.

Maybe in that case the program could be added as an attached text file, with warnings in comments (in that file) that it is not to be used

for generating terms as it is only a conjectural interpretation?

Moreover, I think it is a good practice to include in the beginning of the b-file, if not the whole program, at least a mention which

of the given programs (in the Mathematica/Maple/Program-section, or given as an attached text file) was used to generate it.

Best regards,

Antti

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[M F Hasler]

I think you should add the conjectured equality as a commend and/or formula.

Concerning the program, nothing says that each function given the PROGRAM section must be the function  n -> a(n).

(There are thousands of functions in the PROGRAM sections that do not give a(n): some are otherwise unused "helper functions", some are (sooner or later obsolete) stale copies of other Axxx(n) functions that should be defined and maintained in the appropriate place, yet others plot or draw or print illustrations or other related information.)

So you can add a function "Wouter(n)" corresponding to "your" definition and say in a comment that this is provided for illustration of the conjecture and conjecturally equal to a(n), but it should not be assumed to give a(n).

There are lots of programs given "for illustration only" carrying the mention that they should not be used to compute a(n), even if they should theoretically give the correct values. Usually a comment should say up to where it can safely be used. (Possible reasons include rounding errors, memory overflow, etc.) You can add a comment that it has been checked that your program gives the correct data up to index N.

I think it's perfectly OK to provide such a function for illustration of a conjecture, as long as it is very clearly stated that it should

* not be assumed to give correct terms beyond some limit, and consequently

* not be used to compute more terms of the sequence.

Giving a name different from a(n) or Axxx(n) would help to insist on this.

On Sun, Aug 31, 2025 at 2:29 PM 'Wouter Meeussen' via SeqFan <seq...@googlegroups.com> wrote: 

1/   since it's only a conjecture, and since the editorial consensus is 
(quote)

Programs base on conjecture are disallowed, so this is not ok.
I don't think ok in comments or anywhere else either.
You need to prove the conjecture first. (Until then the lack of a program is not an issue)

(end quote)

What makes you think this is editorial consensus? It's not even correct English...

Conjectured equalities are ABSOLUTELY okay in comments and even formulas (with the mention "conjectured"), this is one of the ways how OEIS contributes to progress in mathematics.

(Of course I mean *real* conjectures, not just wild guesses based on a few initial terms, as we see sometimes in this mailing list...)

Also, the yellow explanation text on the editing page explicitly allows conjectured formulas:

Important: say if you can prove what you entered or if it is only an empirical observation.
Examples:
G.f.: (3/4)*x^3/((1-x)*(1+2*x+x^3)) (conjectured).

> ps. I once added a conjecture to a sequence and later found that D. Knuth
> completed the sequence with it's (short) proof. That's a rare occurrence.

Not that rare. There are many instances where conjectures are given in comments and proved (sometimes much) later. That's exactly how (this part of) OEIS works.

- Maximilian

[A Howroyd]

In the case of A309097,

The conjectured formula was already entered into oeis. If this conjecture is true, then a(n) = (n^2 + 3n)/2 - A006590(n) for n > 0.

So, if the question is: should a program be given to demonstrate this conjecture?, then the answer is no. This is long standing practice to make it harder for others to generate more terms for a sequence based on an unproven result. This rule presumably came about because that is exactly what happened on some occasions. 

On the other hand, the conjecture itself is a valuable contribution to oeis, for the reasons given in the question. Further conjectures, based on a conjecture can also be given in comments. 

[Amarnath Krishnamurthy]

I have thousands of such sequences which have occurred on a different combinatorial idea.

Most are identical but difficult to  correlate. Most can be proved by induction but some are really difficult.

I invite seq fans to visit my website

https://www.murthymaths.in

I have generalised 100s of  results . There are 2200 results uploaded is sets of 100 each.  over 2500 more results are ready to be uploaded. PLease go thru the  details and the sample results.

 Alone I am not able to cope up with the volume of work need  collaboration .

Thanks 

warm regards

Amarnath Murthy

whatsapp number 9969222020

amarna...@gmail.com

Ahmedabad , Gujarat 

India

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[D. S. McNeil]

WM wrote:
>  ps. I once added a conjecture to a sequence and later found that D. Knuth completed the sequence with it's (short) proof. That's a rare occurrence.

Nice, the man himself!

I regularly read the arXiv number theory section to see what's new.  You can imagine my surprise when a few years ago I found https://arxiv.org/abs/2208.03788, "On a conjecture of McNeil", based on a wild guess I'd made so long ago I'd entirely forgotten about it!

Doug

[Kevin Ryde]

A Howroyd <ahow...@hotmail.com> writes:
>
> In the case of A309097,

I uploaded my argument for why the conjecture is right.
Barring any dreadful mistake on my part, the door is
open to new formula.

An individual term goes faster by chasing through existing
knowledge to make a sum up to sqrt(n) instead of n.
The g.f. or a vector of terms have an option in partial
sums of number of divisors style.

> So, if the question is: should a program be given to demonstrate this
> conjecture?, then the answer is no.

Yeah, very little point loading up the prog sections
with "don't use this" material.