[seqfan] Re: Erased copies

[Neil Sloane]

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.
>>
>>
>> à+
>> É.
>> Catapulté de mon aPhone
>>
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>