[seqfan] Re: R: Re: Self-stuffable numbers

[M. F. Hasler]

On Fri, Dec 21, 2018 at 7:22 AM Mason John <john.mason at lispa.it> wrote:

> Hi
> I had decided to stop calculating new values until some optimisation had
> been found.

However as there has been renewed interest, I have calculated values
> through 10^18 and I have posted a new b-file.
> Here are the values post a(84) of the previous file. It took my PC about 8
> hours.
>

Al of these are given in terms of the roots in the b-file
https://oeis.org/A322002/b322002.txt
and the larger roots [1115008, 112621104, 1031111008, 10311111008,
103111111008]
(not in the b-file because it stops at 10^6) ;
among these only the first two come in addition to the infinite sequence
928*R(k), k>=3.

The main challenge is to find larger (primitive) roots, i.e.,
terms of A322323 (without repetitions and) with trailing zeros removed.
Arbitrarily many larger terms of A322323 are then computed
straightforwardly.

- Maximilian


>
> -----Messaggio originale-----
> Da: SeqFan <seqfan-bounces at list.seqfan.eu> Per conto di Neil Sloane
> Inviato: giovedì 20 dicembre 2018 14:32
> A: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Oggetto: [seqfan] Re: Self-stuffable numbers
>
> I'm following this discussion with great interest.
>
> Looks like a pot of bouillabaisse !
>
> When the soup is cooked, I hope you (Maximilian) will write up a report,
> maybe put it on the arXiv, or at least add it to the main sequence A322323.
>
>
> Best regards
> Neil
>
>
>
> On Wed, Dec 19, 2018 at 11:01 PM M. F. Hasler <oeis at hasler.fr> wrote:
>
> > On Wed, Dec 19, 2018 at 9:23 AM Hans Havermann <gladhobo at bell.net>
> wrote:
> >
> > > https://oeis.org/A322323
> > > The appearance of 103008000000 at digit-size 12 was an unexpected
> > delight.
> > > At digit-size 13 we have 1031008000000; at 14, 10311008000000; at
> > > 15, 103111008000000; and yes, the pattern continues!
> > >
> > > Would understanding why this particular infinite family of
> > > self-stuffable numbers exists allow one to find (or rule out the
> > > existence of) other, larger examples?
> >(...)
> > We can understand a little more why & how this works by noticing that
> > x(k) = 928 * R(k+3) * 10^6,  i.e., x(0) = 111*(...), x(1) = 1111*(...)
> > etc and  S(x(k)) = 4279 * 928/4 * R(2(k+3)) * 10^12.
> >
> > From here we see that
> > S(x(k)) / x(k) = R(2(k+3))/R(k+3) * 10^6 * 11*389/4 (and of course
> > R(2(k+3))/R(k+3) = 10^(k+3) + 1).
> (...)