[seqfan] Re: A327265 and A309981

[M. F. Hasler]

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 via SeqFan <
seqfan at list.seqfan.eu> wrote:

> 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.
>
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> Seqfan Mailing list - http://list.seqfan.eu/
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