[seqfan] Re: Bag of digits

[Neil Sloane]

Jonathan:  very interesting construction!  certainly add those sequences to
the OEIS!


Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Fri, Dec 13, 2019 at 1:11 PM jnthn stdhr <jstdhr at gmail.com> wrote:

> Hi folks.
>
> Start with an empty bag of digits.  For each k =0,1,2,... compare the
> digits of k with the contents of the bag.  If a digit of k matches a digit
> in the bag throw them in the trash (they cancel each other out).  Add the
> concatenation m of what remains of k to the sequence and toss the digits of
> m into the bag.  Note that if m contains leading zeros they are stripped
> away in the seq. but added to the bag.
>
> Initially no digits are in the bag so the first ten terms are
> [0,1,2,3,4,5,6,7,8,9] and the bag contains (0,1,2,3,4,5,6,7,8,9).
>
> Now with k=10, we match both digits and nothing remains to add to the
> sequence or the bag,so the sequence remains unchanged and the bag now
> contains (2,3,4,5,6,7,8,9).
>
> With k=11, we match nothing, so 11 is added to the seq. ->
> [0,1,2,3,4,5,6,7,8,9,11]  and two ones are added to the bag ->
> (1,1,2,3,4,5,6,7,8,9).
>
> With k=12 and k=13, all digits are matched and the seq. is unchanged, and
> the bag is left with (4,5,6,7,8,9)
>
> With k=14, the fours cancel and the one remains, so 1 is added to the seq.
> and to the bag.
>
> Etc.
>
> The sequence begins:
> [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 1, 1, 1, 20, 1, 22, 3, 4, 25, 6, 27, 8,
> 29, 3, 33, 3, 3, 40, 1, 42, 3, 44, 5, 6, 47, 8, 49, 5, 5, 55, 5, 60, 1, 62,
> 3, 64, 5, 66, 7, 8, 69, 7, 7, 7, 77, 80, 1, 82, 3, 84, 5, 86, 7, 88, 9, 9,
> 9, 9, 9, 99, 100, 1, 2, 103, 4, 105, 6, 107, 8, 10, 1, 11, 11, 11, 11, 11,
> 20, 1, 22, 13, 4, 125, 6, 127, 8, 129, 131, 33, 1, 13, 13, 140, 1, 42, 13,
> 44, 15, 6, 147, 8, 149, 151, 5, 1, 55, 1, 15, 160, 1, 62, 13, 64, 15, 66,
> 17, 8, 169, 171, 7, 1, 7, 1, 77, 1, 180, 1, 82, 13, 84, 15, 86, 17, 88, 19,
> 191, 9, 1, 9, 1, 9, 1, 99, 200]
>
> The bag size always remains small.  For k > 9 record bag sizes occur at
> (k:record size) 808:11, 8008:12, 10008:13, 80008:15, 800008:16,
> 1000008:17.  Is there a simple explanation for why these k's all end with
> an eight?
>
> The seq. of integers where no digits are canceled out is:
> [0,1,2,3,4,5,6,7,8,9,11,20,22,25,27,29,33,40,42,44,47,49,55,60,...]
>
> The seq. of integers where all digits are canceled out is:
> [10, 12, 13, 15, 17, 19, 30, 32, 34, 35, 37, 39, 50, 52, 54, 56, 57, 59,
> 70, 72, 74, 76, 78, 79, 90, 92, 94, 96, 98, 112, 114, 116, 118, 130, 132,
> 135, 137, 139, 150, 152, 157, 159, 170, 172, 179, 190, 192, 210, 213, 215,
> 217, 219, 230, 235, 237, 239, 250, 257, 259, 270, 279, 290, 310,...]
>
> Should I add these?
>
> -Jonathan
>
> Python code:
>
> seq = []
> bag = ""
> #for first 10K term set range(11256)
> for k in range(1000):
>   m = str(k)
>   for digit in m:
>     if digit in bag:
>       #Remove digit from m and bag.
>       mndx = m.index(digit)
>       m = m[:mndx] + m[mndx+1:]
>       bndx = bag.index(digit)
>       bag = bag[:bndx] + bag[bndx+1:]
>   if m:
>     seq.append(int(m))
>     bag = bag + m
>
> print(seq)
>
> --
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>