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

[M. F. Hasler]

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