A033665(196)

[Allan Wechsler]

The sequence oeis.org/A033665 studies the process of adding a number to its own reverse, and asks how many steps are needed to get from n to a palindrome. Of course this is very decimal-centric: oeis.org/A066057 looks at the same question in binary, and there may be other bases considered -- I haven't checked yet.

The sequence's definition anticipates that with some starting numbers, the process never produces a palindrome, by saying that if it never does, the sequence will report this using the standard -1 convention.

A033665(196) is the first entry that might be -1. It has been run for millions of steps without palindromicity. So far, the collective judgement is that the evidence is not strong enough to put a -1 here, allowing a modest extension of this sequence to n=294. (295 is the next problematic case.) Here the philosophy seems to be, "We don't display entries unless they are proved correct.".

However, this philosophy is not followed with complete consistency. The sequence oeis.org/A023108 is titled, "Numbers that apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed)." If somebody were to run 196 for another trillion steps and find a palindrome, 196 (the first entry) would have to be removed from the sequence.

It's probably necessary to clarify, at this point, that a fairly strong heuristic argument suggests that it is overwhelmingly likely that A033665(196) = -1. But we don't currently have a proof.

For some reason the authors of A066057 (the binary case) made the opposite call, and include -1s at all the positions where we have looked far enough to be heuristically certain that the relevant claim is true.

It is not necessarily the case that multiple standards of certainty are being applied. The heuristic arguments are stronger in the binary case than in the decimal case. But if a consistent standard is being applied, the standard isn't "mathematical certainty". Has this issue been addressed in any more formal way?

Although I don't think I could do it, it seems to me that proving a -1 entry is not entirely out of reach, especially in the binary case. I think it would be possible to "engineer" binary numbers with some property that (1) guaranteed that its successor in the reverse-and-add process is not palindromic, and (2) also guaranteed that the successor had that property. I'm wondering whether anybody has done any of that sort of work.

-- Allan

[Sean A. Irvine]

Hi,

Here is my opinion on Allan's observations that the OEIS is inconsistent in how it deals with conjectural values in Data,  but I'm certainly not the sole authority on this.

- We don't, as a general rule, want conjectural values to appear in Data or b-files.

- We make exceptions for sequences corresponding to published lists (especially in a peer-reviewed publication).

- We might also make exceptions when the data would otherwise be too sparse to justify a sequence.

- We might also make exceptions for sequences having the sole purpose of exploring such unknown values.

- We might also make exceptions under well-established conjectures.

- We definitely do not want to add a mixture of provable "-1's" and conjectural "-1's" within a single sequence.

I also think putting in -1's actually discourages people from trying harder to find the value or a proof.

Sean.

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[Dave Consiglio]

I agree with Sean about the "-1" here. I would prefer to see something like "a(7) > 10^9" and then list a(8) in the notes. This always makes me think that the missing term is attainable and also gives searchers a starting point. 

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[Allan Wechsler]

Sean's list of pragmata about how to decide to include conjectural values was very well-considered and thought-provoking, and goes a long way to answering my questions.

But I had not noticed, until now, that for the binary case, some researchers are claiming actual proofs that some trajectories contain no palindromes. A short note attached to oeis.org/A058042 purports to prove that 22 (10110) has no palindromic iterates. I have not read the note carefully enough to have an opinion on its validity.

If the note does give a valid proof that the trajectory of 22 never hits a palindrome, then we run into Sean's final pragma: we don't want a mixture of conjectural and proved -1's. If A066057(22) = -1 is indeed proved, then I hope that all  the -1's in A066057 have been proved, but the OEIS page itself does not make such a claim.

-- Allan

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[Christopher Landauer]

use -2 for conjectures -1 for proofs? or leave -1 for conjectures, since there are lots of them, and ! or some other symbol for proofs (yes i know that isn't a number, but it seems to me that assigning a non-numerical meaning to a number is fundamentally wrong), or maybe even use ? for conjectures (same conment)

Sent from my iPhone

On Mar 15, 2026, at 19:57, Allan Wechsler <acw...@gmail.com> wrote:



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[M F Hasler]

On Monday, March 16, 2026 at 1:49:57 AM UTC-4 topc...@gmail.com wrote:

use -2 for conjectures -1 for proofs? or (...) 

and ! or some other symbol for proofs (..), or maybe even use ? for conjectures

I think it's a nice idea to use other negative numbers to encode more information about the status of the term, as long as it's not proved.

So the definition could be:

a(n) = ..., or -1 if no such number exists, or  -N < -1 if nonexistence isn't proven yet but N is the largest number for which it is known that a(n) > 10^N if it exists.

(I inserted the slightly clumsy "largest...known..." to discourage people from replacing painfully calculated search limits with "-2", given that it's certainly known the value is > 100 if it exists.)

An apparent drawback is of course that these negative values can and probably will change over time, 

but one has to bear in mind that these aren't terms of the sequence, but just placeholders for unknown/unproven terms,

providing additional information about the current status of (re)search.

If disapproved in the DATA section and in b-files, this quite natural convention could still be used in a-files

(where of course one could also use "?" followed by a comment "# searched up to 10^9", but that would make it more complicated to use the a-file instead of the b-file in calculations that would need more data).

-Maximilian

[Charles Greathouse]

Generally speaking, “as far as we can tell” values belong in an a-file, not the sequence terms or b-file.

Sean, another longstanding category of conjectural terms allowed in the OEIS are terms corresponding to large probable primes.

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[Sean A. Irvine]

I think it was Lenstra that told me long ago that if you find a discrepancy between a large probable prime and a proof of primality, then it is much more likely that the cause is a bug in your implementation of the proof algorithm.

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