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

[Zach DeStefano]

Christian,

Your proof makes sense to me. I independently implemented that closure
check in C++ and verified the number of possible ADCs for bases 2 to 10.
Going further than that requires more computing resources than I currently
have or a more space-efficient algorithm.

I don't know if there will be a base b where there does not exist a k such
that the closure of the set of unique ADCs found between b^k and b^(k + 1)
is not contained in the set of all found ADCs for b, but assuming that is
always true, verifying additional terms just requires more compute.

Allan,

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

It seems that bases 2 and 3 are the only ones where we can get the ADC
corresponding to {0, 1} from what I've checked. (at least up to base 10). I
wonder what makes this potentially unreachable for all other bases.

All,

The question remains, is the sequence of unique ADCs: 3, 6, 10, 29, 24, 84,
91, 179, 121, ... interesting enough to be its own entry in the OEIS?

- Zach

On Sat, Apr 8, 2023 at 9:19 AM M. F. Hasler <oeis at hasler.fr> wrote:

> Yes, the sum of the powers of two of the elements is the *most natural*
> encoding for a finite set of non-negative integers that may possibly exist.
>  (It is widely used (as "bitmask"), for example in PARI 's vecexract()
> command, and that might be the least popular example ... Also the main
> reason of existence of the "bitfield / bitset" structure and similar data
> types).
>
> In the present case, where we are dealing with digits in the first place,
> it might also be natural to simply concatenate the possible digit outcomes
> in increasing order except for zero that must of course come last. So, for
> example a(n) = 249 means that the possible outcome is 2 or 4 or 9.
>
> - Maximilian
>
>
> On Wed, Apr 5, 2023, 23:40 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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> > >
> >
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>
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