Two identical digits

 

There are at least 2 identical digits in the result of a(n) * a(n+1). This is the lexicographically earliest sequence of distinct positive terms with this property (I hope).

S = 1,11,2,22,3,33,10,20,5,23,7,16,9,13,17,15,35,19,6,24,12,21,25,4,28,8,...

Examples

 1 * 11 =  11

11 *  2 =  22

 2 * 22 =  44

22 *  3 =  66

 3 * 33 =  99

33 * 10 = 330

10 * 20 = 200

20 *  5 = 100

 5 * 23 = 115

 7 * 16 = 112

Etc.

____________________

Around midnight update

> Giorgos Kalogeropoulos was quick to compute S; here are the first 100 terms:

S = 1, 11, 2, 22, 3, 33, 10, 20, 5, 23, 7, 16, 9, 13, 17, 15, 35, 19, 6, 24, 12, 21, 25, 4, 28, 8, 14, 18, 29, 31, 32, 34, 26, 38, 30, 36, 37, 27, 40, 45, 39, 41, 42, 44, 46, 47, 43, 52, 50, 48, 49, 51, 59, 56, 55, 57, 58, 54, 53, 61, 60, 65, 62, 67, 63, 68, 66, 64, 69, 71, 72, 70, 73, 75, 74, 78, 77, 76, 79, 81, 82, 80, 83, 85, 89, 84, 86, 90, 92, 87, 88, 91, 94, 97, 100, 93, 95, 101, 96, 102, ...

> I didn't send a plot because it soon becomes a straight line. I will try some variations.

GK

How about T: Every digit has a twin (first 100 terms hereunder and graph of the first 1000 terms):

T = 1, 11, 2, 22, 3, 33, 34, 66, 17, 101, 10, 110, 20, 55, 21, 132, 9, 202, 5, 220, 15, 77, 13, 88, 24, 176, 12, 99, 19, 121, 28, 143, 7, 165, 27, 44, 25, 209, 16, 187, 6, 198, 14, 242, 23, 303, 4, 275, 8, 264, 32, 3159, 154, 18, 253, 29, 231, 26, 297, 364, 286, 31, 319, 314, 415, 294, 429, 238, 572, 182, 561, 200, 583, 190, 739, 155, 658, 246, 407, 276, 363, 277, 578, 279, 390, 295, 374, 273, 385, 288, 351, 308, 338, 461, 333, 341, 339, 298, 396, 278,  ...

First 1000 terms of T

First 3000 terms of T

GK

This sequence is finite: the largest product of 2 successive terms is 99887766554433221100. So, the divisors of this products (namely the terms of this sequence) are finite. If we want another variation with infinite terms (sequence U hereunder), we should ask that "We have at least 2 digits of every digit in a(n) * a(n+1)". The first 100 terms of U are:

U = 1, 11, 2, 22, 3, 33, 34, 66, 17, 101, 10, 110, 20, 55, 21, 37, 6, 74, 9, 111, 4, 222, 50, 44, 25, 88, 13, 77, 15, 202, 5, 220, 30, 303, 7, 143, 8, 264, 16, 132, 32, 198, 14, 242, 23, 99, 12, 176, 19, 121, 28, 275, 24, 187, 18, 154, 26, 231, 29, 253, 35, 165, 27, 286, 31, 319, 38, 307, 76, 579, 57, 193, 89, 199, 56, 377, 112, 387, 178, 471, 358, 59, 274, 73, 137, 87, 153, 174, 209, 42, 481, 63, 497, 126, 344, 71, 141, 93, 282, 142,  ...

Graph of the first 3000 terms of U

ÉA

> Many thanks, Giorgos, good job and interesting variations!

____________________

Next day update

Jean-Marc Falcoz proposed another variation leading to a finite sequence (X, first 113 terms below):

"Lexicographically earliest sequence of distinct positive terms such that a(n)*a(n+1) contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present)... 9 digits 9 (if 9 is present)." The sequence is finite as the largest possible product a(n)*a(n+1) is 999999999888888887777777666666555554444333221.

X = {1,22,202,130033,1942165,285373,15637242,2715027,1235905,28677249,1832656,13337667,1840302,13218127,32769805,13257151,12564203,2740608,9287193,778209,8253074,2972871,1270532,58383242,4563358,30184653,110706148,3120274,23859178,18217997,11831885,19654883,73038,45748,772824,30324531,536086,1028854,647579,687569,2099505,12067389,3691397,1751712,10406299,5316246,6479112,857237,4837218,321163,13561818,3335508,7056003,6427485,5046197,12810922,40711,103494,41881,103224,32206,131439,16306,252939,215286,8005062,6913557,4849085,7099371,4863044,503106,6389,4906,135887,38659,91967,386565,117585,208823,2181485,10746603,319308761,163341483,22795339,14571196,10597864,22884946,19341827,11718979,3708117,9318866,7142667,7365968,11293599,4635675,6999079,647736,7203809,3397886,7223769,3120747,8176448,27845437,19947733,22826425,10582189,6091997,12214841,1992294,11823328,37926571,34974516,66380809,...}

We have indeed:

               1 * 22 = 22

             22 * 202 = 4444

         202 * 130033 = 26266666

     130033 * 1942165 = 252545541445

     1942165 * 285373 = 554241452545

    285373 * 15637242 = 4462446661266

   15637242 * 2715027 = 42455534235534

    2715027 * 1235905 = 3355515444435

   1235905 * 28677249 = 35442355425345

   28677249 * 1832656 = 52555532443344

   1832656 * 13337667 = 24443355453552

   13337667 * 1840302 = 24545335255434

   1840302 * 13218127 = 24325345554354

  13218127 * 32769805 = 433155444255235

  32769805 * 13257151 = 434434253125555

  13257151 * 12564203 = 166565536365653

   12564203 * 2740608 = 34433555255424

    2740608 * 9287193 = 25452555433344

     9287193 * 778209 = 7227377177337

     778209 * 8253074 = 6422616464466

    8253074 * 2972871 = 24535324355454

    2972871 * 1270532 = 3777127737372

   1270532 * 58383242 = 74177777224744

   58383242 * 4563358 = 266423634446636

   4563358 * 30184653 = 137743377744774

 30184653 * 110706148 = 3341626662346644

  110706148 * 3120274 = 345433515244552

   3120274 * 23859178 = 74447172774772

  23859178 * 18217997 = 434666433226466

  18217997 * 11831885 = 215553245434345

  11831885 * 19654883 = 232554315344455

     19654883 * 73038 = 1435553344554

        73038 * 45748 = 3341342424

       45748 * 772824 = 35355152352

    772824 * 30324531 = 23435525345544

    30324531 * 536086 = 16256556525666

     536086 * 1028854 = 551554225444

     1028854 * 647579 = 666264244466

      647579 * 687569 = 445255245451

     687569 * 2099505 = 1443554553345

   2099505 * 12067389 = 25335543542445

   12067389 * 3691397 = 44545523552433

    3691397 * 1751712 = 6466264421664

   1751712 * 10406299 = 18228838833888

   10406299 * 5316246 = 55322445433554

    5316246 * 6479112 = 34444553253552

     6479112 * 857237 = 5554134533544

     857237 * 4837218 = 4146642246666

     4837218 * 321163 = 1553535444534

    321163 * 13561818 = 4355554154334

   13561818 * 3335508 = 45235552433544

    3335508 * 7056003 = 23535354454524

    7056003 * 6427485 = 45352353442455

    6427485 * 5046197 = 32434355524545

   5046197 * 12810922 = 64646436163634

     12810922 * 40711 = 521545445542

       40711 * 103494 = 4213344234

       103494 * 41881 = 4334432214

       41881 * 103224 = 4323124344

       103224 * 32206 = 3324432144

       32206 * 131439 = 4233124434

       131439 * 16306 = 2143244334

       16306 * 252939 = 4124423334

      252939 * 215286 = 54454225554

     215286 * 8005062 = 1723377777732

    8005062 * 6913557 = 55343452425534

    6913557 * 4849085 = 33524425545345

    4849085 * 7099371 = 34425453425535

    7099371 * 4863044 = 34524553545324

     4863044 * 503106 = 2446626614664

        503106 * 6389 = 3214344234

          6389 * 4906 = 31344434

        4906 * 135887 = 666661622

       135887 * 38659 = 5253255533

        38659 * 91967 = 3555352253

       91967 * 386565 = 35551223355

      386565 * 117585 = 45454245525

      117585 * 208823 = 24554452455

     208823 * 2181485 = 455544242155

   2181485 * 10746603 = 23443553245455

 10746603 * 319308761 = 3431484488888883

319308761 * 163341483 = 52156366556632563

 163341483 * 22795339 = 3723424477747737

  22795339 * 14571196 = 332155352455444

  14571196 * 10597864 = 154423553525344

  10597864 * 22884946 = 242531545355344

  22884946 * 19341827 = 442636666436342

  19341827 * 11718979 = 226666464434633

   11718979 * 3708117 = 43455345252543

    3708117 * 9318866 = 34555445435322

    9318866 * 7142667 = 66561556655622

    7142667 * 7365968 = 52612656556656

   7365968 * 11293599 = 83188288838832

   11293599 * 4635675 = 52353454544325

    4635675 * 6999079 = 32445455543325

     6999079 * 647736 = 4533555435144

     647736 * 7203809 = 4666166426424

    7203809 * 3397886 = 24477721747774

    3397886 * 7223769 = 24545543552334

    7223769 * 3120747 = 22543555435443

    3120747 * 8176448 = 25516625566656

   8176448 * 27845437 = 227676767667776

  27845437 * 19947733 = 555453342544321

  19947733 * 22826425 = 455335431244525

  22826425 * 10582189 = 241553543544325

   10582189 * 6091997 = 64466663641433

   6091997 * 12214841 = 74412774727477

   12214841 * 1992294 = 24335554435254

   1992294 * 11823328 = 23555545434432

  11823328 * 37926571 = 448418288848288

  37926571 * 34974516 = 1326463464264636

  34974516 * 66380809 = 2321636666463444

Merci Jean-Marc ! We love those "monsters"!

(pix taken from here)