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

[Michael Branicky]

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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> >
>
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