[seqfan] Erased copies

[Éric Angelini]

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