[seqfan] Re: Re Another Split & Multiply sequence from Eric A.

[Allan Wechsler]

Zach, Michael Branicky privately mailed me the same number, so we can be
confident that only 121 digit achievability patterns occur.

I am convinced by Neil and Tom that while these codings are useful behind
the scenes, they rarely have any business appearing in sequences unless the
actual values turn out to have interesting mathematical properties.

It is interesting, isn't it, that from no starting number can we achieve 0
and 1 only.

I think the number of different achievability patterns for each base that
Zach discusses would make a good sequence, but we need a toolkit for
proving which achievability patterns are impossible. Otherwise we would
never be sure that there wasn't a million-digit counterexample.

On Thu, Apr 6, 2023 at 11:07 AM Zach DeStefano <zachdestefano at gmail.com>
wrote:

> To generate values in A361340 I also used a similar binary encoding. To
> answer your ADC questions about which values are achievable, it empirically
> seems like the base of 2 is the only one that generates all ADCs (ignoring
> 0). For base 10, (in the window that I checked), we only see 121 of the
> 1023 unique ADCs produced. 34119 is the last number (at least up to 2^28)
> which produces a new ADC.
>
> For base 3, the only ADC not produced is 5. I suspect for a small case like
> this, it will be easy to prove that such an ADC is impossible, but for
> larger bases with more missing ADCs, this becomes much trickier.
>
> Assuming there are no large ADC surprises, the sequence of the number of
> unique ADCs for each base (starting with 2) is: 3, 6, 10, 29, 24, 84, 91,
> 179, 121, 620, 284...
>
> One interesting observation here is that there is an inverse relationship
> between the sizes of the terms of A361340 and this sequence. That is to
> say, prime bases appear to cover relatively more ADCs and reach the
> maximal ADC the fastest while bases with many prime factors cover very few
> ADCs and take the longest to reach the maximal ADC.
>
> I don't know if this sequence is particularly interesting or useful beyond
> that however.
>
> - Zach
>
>
> On Thu, Apr 6, 2023 at 8:33 AM Michael Branicky <branicky at gmail.com>
> wrote:
>
> > Yes, Allan!  To generate the larger values in A361340 I needed to switch
> to
> > a binary encoding instead of a set as in my posted code to save memory
> (see
> > alternate linked program -- currently in draft edits -- which could be
> > easily altered to answer your ADC questions)
> >
> > I think some of Michael De Vlieger's images at A361338 also explore this
> > notion nicely.
> >
> > On Wed, Apr 5, 2023 at 10:39 PM Allan Wechsler <acwacw at gmail.com> wrote:
> >
> > > The number of digits achievable by playing the SMI game starting from n
> > is
> > > now A361338. But this sequence does not tell one precisely which digits
> > can
> > > be obtained. Of course we could imagine creating ten *more* sequences,
> > for
> > > each d from 0 to 9, "Numbers from which it is possible to obtain d."
> But
> > we
> > > could encode the achievable digits as a binary number. For example,
> from
> > > 127 it is possible to obtain either 6 or 4; we could encode this as
> 2^6 +
> > > 2^4 = 80. These achievable digit codes will run from 1 to 1023; 0 is
> > > impossible because every starting n can produce *some* digit. (Maybe
> you
> > > could construct an argument that you can't obtain anything starting
> from
> > 0,
> > > depending on exactly how the definition is lawyered.) There are
> probably
> > > other unachievable codes; is there a starting number from which only 3
> > and
> > > 7 can be obtained?
> > >
> > > Anyway, this "achievable digit code" (ADC) sequence will start (from
> > n=1):
> > > 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 2, 4, 8, 16, 32, 64, 128, 256,
> > 512,
> > > 1, ...
> > >
> > > The values of  ADC(n) will be just powers of 2 up to ADC(111) = 2. Then
> > > ADC(112) = 10 is the first deviation from that behavior. Then ADC(113)
> =
> > > 516 (that is, 2^9 + 2^2, because 9 and 2 are the digits obtainable from
> > > 113).
> > >
> > > What values does ADC(n) take? How many values are unachievable? What is
> > the
> > > smallest unachievable value?
> > >
> > > The binary encoding is rather unnatural, but it does capture in a
> single
> > > integer value a summary of the possible outcomes from a given starting
> > > value. Then A361338 = A000120(ADC(n)), the binary weight of the
> > achievable
> > > digit code. I have a vague hope that the graph of ADC(n) will suggest
> > more
> > > questions and maybe even some answers.
> > >
> > >
> > >
> > > On Wed, Apr 5, 2023 at 10:48 PM Neil Sloane <njasloane at gmail.com>
> wrote:
> > >
> > > > (the previous thread had gotten too knotted - gmail does some things
> > > well,
> > > > but it can make a bowl of spaghetti out of a bunch of emails that are
> > > > closely related in time)
> > > >
> > > > I just added A361341-A361349 for the numbers with 2 through 10
> > > single-digit
> > > > children.
> > > >
> > > > The full set is A361337-A361349, with A361338 the central one.
> > > > That's probably enough sequences for now. What we need next are some
> > > > theorems.
> > > >
> > > > Best regards
> > > > Neil
> > > >
> > > > Neil J. A. Sloane, Chairman, OEIS Foundation.
> > > > Also Visiting Scientist, Math. Dept., Rutgers University,
> > > > Email: njasloane at gmail.com
> > > >
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