[seqfan] Re: A327265 and A309981

[Peter Munn]

On Sat, April 1, 2023 2:13 pm, M. F. Hasler 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 .
>
[...]
> 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,

This does not quite follow, because there could be a sequence of such m,
each requiring a longer signature to distinguish m from n.

Nevertheless, the idea of equivalent sequences that use the characteristic
function of primes instead of tau seems worth pursuing. The proof of the
existence of the equivalent signature for n would come from considering
the a prime p > n and the residues of subsequent primes modulo p.

Best regards,

Peter

> which would mean that prime gaps become
> periodic (or something like that).
>
> - Maximilian
>