[seqfan] Re: Erased copies

Thu Oct 25 01:55:08 CEST 2018

On the question of 0 vs -1 for the empty string, it depends on whether we
are working in the realm of numbers or of strings (as in "combinatorics on
words").  In the latter case we definitely want to distinguish between the
string 0 and the empty string \epsilon.  So we need to have both versions.

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 Wed, Oct 24, 2018 at 6:46 PM M. F. Hasler <seqfan at hasler.fr> wrote:

> On Wed, Oct 24, 2018 at 5:24 PM Allan Wechsler <acwacw at gmail.com> wrote:
>
> > Under the "delete multiple digits" operation, does 10201 map to 20, or
> does
> > it map to 2? Either answer makes sense, depending on the detailed
> > definition of the map.
> >
>
> IMHO, "Delete all digits that appear more than once." or even better :
> "Keep only digits that appear only once."
> would be less ambiguous.
>
> Also, many might prefer the variant using 0's over that with -1's,
> so I'd be against judging one "better" than the other, which is a matter of
> taste.
> It makes perfectly sense to say the empty "string" (i.e., sequence of
> digits) is the decimal representation of 0.
> (This is the convention used by many mathematicians and also computer
> algebra systems.)
>
> - Maximilian
>
>
>
> >
> > On Wed, Oct 24, 2018 at 12:37 PM Neil Sloane <njasloane at gmail.com>
> wrote:
> >
> > > Correction: A320487 is
> > >
> > > a(0)= 1; thereafter a(n) is obtained by applying the "delete multiple
> > > digits" map m -> A320485(m) to 2*a(n-1).
> > > Best regards
> > > Neil
> > >
> > >
> > >
> > >
> > >
> > > On Wed, Oct 24, 2018 at 12:28 PM Neil Sloane <njasloane at gmail.com>
> > wrote:
> > >
> > > > Eric,  That is a lovely idea, one of your best!  I like it a lot.
> > > >
> > > > To start the ball rolling, I created A320485 for the basic operation
> > > > applied
> > > > just once to n itself, writing -1 if the result is the empty string.
> > > >
> > > > And I created A320486 for the basic operation applied
> > > > just once to n itself, writing 0 if the result is the empty string.
> (I
> > > > don't like it as much, but it
> > > > has to be in the OEIS too).
> > > >
> > > > Then A320487 is the doubling sequence, starting with 1, which as you
> > > > observed is periodic with period 28.
> > > >
> > > > Maybe you could send in some others , for example,
> > > > - number of steps before the operation reaches a fixed point
> > > > - the trebling sequence starting at 1
> > > > - the finite "factorial" sequence in your email, which is
> > > > 2 x 1    = 2
> > > > 3 x 2    = 6
> > > > 4 x 6    = 24
> > > > 5 x 24   = 120
> > > > 6 x 120  = 720
> > > > 7 x 720  = (5040) = 54
> > > > 8 x 54   = 432
> > > > 9 x 432  = (3888) = 3
> > > > 10 x 3   = 30
> > > > 11 x 30  = 330
> > > > End. Only 11 terms.
> > > >
> > > > -  the start a(1) = 2 generates 23 terms [last one
> > > > being  23 x 198 = (4554) End]
> > > > - the start a(1) = 3 produces only 12 terms [last one 12 x 407 =
> (4884)
> > > > End].
> > > >
> > > > If you have time, please submit all these and more!
> > > >
> > > > Here is the wording I used for the doubling sequence A320487:
> > > >
> > > > a(0)= 1; thereafter a(n) is obtained by applying the "delete multiple
> > > > digits" map m -> A320485(m) to a(n-1).
> > > >
> > > > At first I called this the Angelini map, but then I realized that you
> > > > would not be able to use that name!
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > On Wed, Oct 24, 2018 at 7:31 AM Éric Angelini <
> eric.angelini at skynet.be
> > >
> > > > wrote:
> > > >
> > > >> Hello SeqFan,
> > > >> Take an integer and keep only its distinct digits in their
> > > >> apparition order. Example:
> > > >>
> > > >> 1231 becomes 23
> > > >> 1123 becomes 23
> > > >> 11231 becomes 23
> > > >> and
> > > >> 11023 becomes 23 too (as we don't accept leading zeroes).
> > > >> Note that 112323 disappears immediately.
> > > >>
> > > >> Now chose a function F, a starting term a(1) and iterate.
> > > >>
> > > >> Say, for instance, that the function F is "double" and a(1) is 19:
> > > >>
> > > >>
> > > >>
> > >
> >
> 19,38,76,152,304,608,(1216),26,52,104,208,416,832,(1664),14,28,56,(112),2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,(65536),3,6,12,24,48,96,192,384,768,1536,3072,(6144),61,(122),1,2,
> > > >> ... (loop).
> > > >>
> > > >> The "triple" function F starting with a(1) = 37 stops immediately,
> > > >> of course (as 37 x 3 --> 111).
> > > >>
> > > >> The "square" function is interesting as some huge integers appear
> > > >> – that sometimes collapse into a 2- or 3-digit integer. I didn't
> > > >> explore thoroughly this domain (fixed points, loops, flights,
> > > >> altitudes, etc.) -- only played a bit with the idea.
> > > >>
> > > >> A sequence I would like to see is the one dealing with the function
> > > >> F = (n+1)*a(n) that would generate, for the smallest a(1), more
> > > >> than 100 terms...
> > > >>
> > > >> For a(1) = 1, we have the sequence:
> > > >>
> > > >> 2 x 1    = 2
> > > >> 3 x 2    = 6
> > > >> 4 x 6    = 24
> > > >> 5 x 24   = 120
> > > >> 6 x 120  = 720
> > > >> 7 x 720  = (5040) = 54
> > > >> 8 x 54   = 432
> > > >> 9 x 432  = (3888) = 3
> > > >> 10 x 3   = 30
> > > >> 11 x 30  = 330
> > > >> End. Only 11 terms.
> > > >>
> > > >> If I'm not wrong, the start a(1) = 2 generates 23 terms [last one
> > > >> being  23 x 198 = (4554) End] and the start a(1) = 3 produces only
> > > >> 12 terms [last one 12 x 407 = (4884) End].
> > > >> What would be the smallest a(1) generating 100 terms or more?
> > > >>
> > > >> P.-S.
> > > >> A friend of mine thinks that no
> > > >> integer < 10000000 generates
> > > >> any 100-term sequence, according
> > > >> to a program he wrote: the longest
> > > >> sequence would have 78 terms
> > > >> and the smallest generating-term
> > > >> would be 19 128.
> > > >>
> > > >> Here is the 19 128 sequence [between
> > > >> brackets are the terms that will
> > > >> be "simplified"] :
> > > >>
> > > >> 19128 38256 [114768] 4768 19072 95360 572160 [4005120] 4512 [36096]
> > 309
> > > >> 2781 27810 [305910] 3591 43092 [560196] 5019 [70266] 702 [10530] 153
> > > [2448]
> > > >> 28 476 [8568] 56 1064 [21280] 180 3780 83160 [1912680] 92680
> [2224320]
> > > 430
> > > >> [10750] 175 [4550] 40 [1080] 18 504 [14616] 4 120 3720 [119040] 94
> > 3102
> > > >> 105468 [3691380] 69180 [2490480] 298 [11026] 26 [988] 9 351 [14040]
> 1
> > 41
> > > >> [1722] 17 731 32164 [1447380] 17380 [799480] 7480 [351560] 3160
> > [151680]
> > > >> 5680 [278320] 7830 [391500] 3915 [199665] 15 780 [41340] 130 [7020]
> 72
> > > 3960
> > > >> [221760] 1760 [100320] 132 [7656] 75 [4425] 25 [1500] 15 915 56730
> > > >> [3573990] 570 36480 [2371200] 371 [24486] 286 [19162] 962 [65416]
> 541
> > > >> [37329] 729 [51030] 513 [36423] 642 [46224] 6 438 [32412] 341
> [25575]
> > 27
> > > >> [2052] 5 385 [30030] 0 --> END.
> > > >>
> > > >> [Note: I've been away a long time from the SeqFan list -- and
> > > >> if all this is old hat, please ignore this
> > > >> and forgive me]
> > > >> Best,
> > > >> É.
>
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