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

[Allan Wechsler]

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/
>