[seqfan] Re: A327265 and A309981

[M. F. Hasler]

On Mon, Apr 3, 2023 at 11:42 AM <hv at crypt.org> wrote:

> I'd be happy to collaborate on such an endeavour.
>
Great :

> I think there is an initial empirical phase: it should be easy to generate
> a list of divisor counts up to somewhere between 1e10 and 1e12, and use
> that as an initial evidence base of signatures that definitely repeat.
>

Yes, good idea.

> I suspect that a proportion of what remains as apparently unique will be
> easy to prove unique; I anticipate that there will remain cases that
> require, for example, the solution to a Pell-like equation in numbers
> of a particular prime signature, whose values rapidly go beyond our
> current ability to factorize. I'm sure some of those will be provable
> on modular grounds, but I doubt that all will be.
>
That will be the interesting part. I wonder where this starts and where we
might get stuck.


> :(Maybe on a wiki page? I actually tentatively started one...)
>


> Is it an OEIS wiki page? Will you share a link?
>

I started https://oeis.org/wiki/A309981
Maybe it should be renamed to "The tau signature"
(if this term finds approval) -- maybe appended or prefixed to the
A-number. (Anyway if we rename the old name will be a redirect to the new
one.)

But it is really just a stub. I was very busy and didn't have the time to
work on this so far
(whence also the late reply -- I'm sorry)

PS: Richard's preprint about tau(n)=tau(n+1) is for me just a confirmation
that this would really be of interest.

PS2: Maybe in order to avoid to flood the list and annoy those not
interested,
we shoud make a private thread for continuing the discussion, on which we
can add anyone upon request ...

-Maximilian

"M. F. Hasler" <oeis at hasler.fr> wrote:
:I like the idea in that/these sequences.
:I propose to call "tau signature" sequences of the form
:t(n,k) = (tau(n), tau(n+1), ..., tau(n+k)), [tau = sigma_0 = numdiv]
:either for any k (in that case /a/ tau signature doesn't always uniquely
:identify n)
:or for the least k* = A309981(n) such that t(n,k*) uniquely identifies n
:(and then let t(n) = t(n,k*))
:and we should call it *the* tau signature of n .
:
:I think it would be valuable to list the tau signatures t(n) for all
:integers, along with an explanation.
:(Maybe on a wiki page? I actually tentatively started one... That makes up
:interesting puzzles! I:-)
:So that would be the table T(n,k) = tau(n+k) with row lengths A309981:
:1: 1
:2: 2,2
:3: 2,3
:4: 3,2
:5: 2,4
:6: 4,2,4
:7: 2,4,3
:...
:Obviously each row starts with the 2nd element of the previous row:
:T(n+1, k=0,1,2...) = T(n, k=1,2,3...)
:(as long as the column index does not exceed the length of the respective
:row)
:
:Do we have a formal proof that any n has a tau signature?
:Yes, because if this weren't the case, it would mean that there's m > n
:such that
:tau(m+k) = tau(n+k) for all k, which would mean that prime gaps become
:periodic (or something like that).
:
:- Maximilian
:
:On Fri, Mar 31, 2023 at 8:16 PM Brendan McKay:
:> There is some discussion of these sequences at
:> https://mathoverflow.net/questions/439067/on-the-oeis-sequence-a327265
:> which someone in this list might be able to help with.
:>
:> Brendan.