Make the average of two successive terms and look

Take any pair of successive terms a(n) and a(n+1) in the sequence S.
Compute A = [a(n) + a(n+1)]/2.
No digit of A is visible in a(n) and no digit of A is visible in a(n+1).

S is the lexicographically earliest sequence of distinct positive terms with this property.

S = 1,3,5,7,2,4,6,8,10,34,11,33,9,20,46,13,31,14,…

Ckeck:

(1+3)/2 = 2 and 2 is not visible in the pair (1,3);

(3+5)/2 = 4 and 4 is not visible in the pair (3,5);

(5+7)/2 = 6 and 6 is not visible in the pair (5,7);

(7+2)/2 = 4.5 and neither 4 nor 5 are visible in the pair (7,2);

(2+4)/2 = 3 and 3 is not visible in the pair (2,4);

(4+6)/2 = 5 and 5 is not visible in the pair (4,6);

(6+8)/2 = 7 and 7 is not visible in the pair (6,8);

(8+10)/2 = 9 and 9 is not visible in the pair (8,10);

(10+34)/2 = 22 and 2 is not visible in the pair (10,34); no integer < 34 could be used as:

         (10+9)/2 = 9.5 and 9 is visible in the pair (10,9)

         (10+11)/2 = 10.5 and 1 at least is visible in the pair (10,11)

         (10+12)/2 = 11 and 1 is visible in the pair (10,12)

         ...

         (10+19)/2 = 14.5 and 1 is visible in the pair (10,19)

         (10+20)/2 = 15 and 1 is visible in the pair (10,20)

         ...

         (10+29)/2 = 19.5 and 1 is visible in the pair (10,29)

         (10+30)/2 = 20 and 0 is visible in the pair (10,30)

         (10+31)/2 = 20.5 and 0 is visible in the pair (10,31)

         (10+32)/2 = 21 and both 2 and 1 are visible in the pair (10,32)

         (10+33)/2 = 21.5 and 1 is visible in the pair (10,33)

         (10+34)/2 = 22 = hit, as no 2 is visible in the pair (10,34).

etc.

Hope someone will have a look, correct the typos, extend and submit S to the OEIS.
Best,
É.

____________________

September 10 update

Leo Broukhis was the first to react (and correct S) on Math-Fun:

> The next value after 34 must be 9, because 9 was not yet used, and (34+9)/2 =21.5 does not share digits with 34 or 9.  

I’ve generated about a million elements. It took me about an hour. The last 2-digit number, 99, appears at position 112, the last 3-digit number, 959, @1261, then 9182 @23922, and 93152 @487999. 

The largest observed number among the million elements is 12894362 @978601.

Leo

(see attached first 1000 terms of S)

S = 1, 3, 5, 7, 2, 4, 6, 8, 10, 34, 9, 20, 42, 18, 30, 14, 31, 13, 35, 17, 33, 11, 37, 15, 39, 16, 38, 50, 22, 40, 26, 41, 19, 36, 52, 24, 46, 21, 45, 27, 44, 28, 58, 70, 23, 57, 25, 47, 29, 51, 73, 43, 60, 82, 12, 48, 62, 32, 56, 84, 49, 61, 83, 55, 71, 53, 75, 91, 63, 80, 54, 72, 90, 64, 81, 59, 77, 92, 69, 85, 101, 65, 89, 66, 88, 100, 67, 93, 76, 94, 79, 97, 74, 96, 200, 68, 86, 102, 87, 103, 341, 78, 106, 338, 107, 337, 95, 225, 407, 98, 204, 99, 203, 620, 126, 474, 104, 340, 110, 334, 111, 333, 115, 339, 105, 347, 109, 335, 113, 331, 114, 330, 118, 336, 108, 343, 116, 368, 117, 363, 131, 313, 135, 309, 136, 308, 137, 307, 138, 306, 139, 305, 147, 303, 141, 304, 140, 310, 134, 311, 133, 315, 143, 301, 144, 300, 154, 370, 119, 361, 130, 314, 186, 318, 163, 317, 167, 344, 160, 384, 148, 366, 149, 367, 193, 316, 168, 346, 164, 380, 158, 374, 150, 378, 506, 162, 444, 146, 364, 180, 354, 170, 319, 161, 349, 166, 348, 171, 353, 175, 345, 179, 357, 181, 351, 173, 355, 177, 359, 185, 369, 191, 360, 184, 350, 174, 358, 190, 379, 145, 375, 153, 371, 157, 381, 151, 373, 155, 377, 159, 385, 169, 386, 502, 176, 393, 501, 165, 383, 505, 187, 400, 156, 388, 500, 172, 428, 178, 406, 188, 356, 198, 412, 189, 365, 183, 401, 265, 409, 222, 404, 226, 408, 224, 402, 220, 442, 218, 448, 152, 454, 208, 422, 192, 414, 196, 411, 194, 376, 508, 244, 416, 199, 415, 197, 419, 182, 418, 242, 420, 202, 424, 206, 429, 205, 427, 209, 425, 207, 455, 211, 449, 195, 417, 249, 421, 245, 475, 125, 481, 129, 471, 142, 458, 214, 446, 221, 445, 215, 451, 227, 405, 229, 441, 219, 447, 260, 410, 256, 484, 122, 478, 128, 472, 201, 465, 281, 426, 240, 462, 212, 464, 210, 456, 284, 450, 216, 460, 217, 459, 277, 440, 262, 480, 247, 469, 241, 466, 248, 470, 246, 461, 285, 507, 255, 477, 124, 476, 290, 452, 270, 492, 121, 479, 127, 485, 261, 486, 228, 482, 250, 488, 112, 489, 251, 487, 613, 267, 494, 252, 490, 264, 491, 266, 496, 258, 522, 254, 498, 604, 430, 600, 223, 577, 231, 569, 233, 567, 235, 565, 237, 563, 239, 561, 253, 555, 257, 509, 259, 495, 271, 499, 272, 504, 274, 497, 269, 531, 283, 517, 275, 457, 279, 521, 287, 511, 289, 519, 293, 515, 291, 523, 297, 525, 295, 467, 294, 712, 268, 512, 276, 510, 278, 514, 280, 518, 282, 468, 286, 528, 292, 516, 298, 530, 302, 520, 362, 526, 288, 532, 296, 538, 382, 550, 332, 556, 120, 546, 854, 390, 552, 328, 560, 322, 558, 396, 566, 236, 572, 320, 562, 238, 582, 232, 568, 312, 576, 392, 570, 326, 622, 329, 503, 325, 557, 263, 537, 299, 513, 731, 387, 533, 389, 535, 273, 527, 395, 539, 397, 571, 321, 559, 323, 573, 391, 529, 399, 553, 327, 593, 703, 213, 587, 701, 230, 592, 700, 403, 627, 372, 590, 352, 536, 864, 542, 718, 394, 611, 398, 612, 848, 342, 668, 132, 669, 431, 614, 436, 602, 432, 603, 433, 606, 437, 607, 423, 608, 434, 609, 493, 617, 439, 616, 438, 662, 800, 234, 766, 123, 677, 324, 676, 443, 601, 843, 541, 705, 547, 709, 543, 713, 463, 637, 413, 631, 473, 628, 483, 618, 830, 524, 702, 534, 704, 540, 722, 544, 708, 554, 710, 548, 724, 598, 728, 594, 730, 900, 243, 767, 901, 435, 777, 545, 707, 549, 711, 575, 717, 551, 715, 583, 737, 585, 735, 589, 733, 591, 729, 597, 723, 599, 721, 945, 581, 739, 903, 453, 747, 579, 741, 595, 725, 907, 720, 578, 742, 580, 744, 588, 732, 904, 610, 834, 584, 738, 1106, 639, 803, 625, 809, 626, 802, 624, 806, 623, 805, 629, 841, 605, 823, 659, 815, 633, 811, 634, 804, 630, 808, 574, 748, 1103, 641, 810, 638, 813, 635, 819, 636, 814, 586, 818, 596, 812, 648, 816, 643, 801, 644, 822, 640, 826, 649, 821, 645, 829, 642, 820, 646, 824, 650, 828, 652, 836, 564, 838, 656, 832, 663, 825, 655, 833, 615, 839, 661, 844, 660, 831, 619, 835, 653, 889, 621, 845, 695, 849, 666, 840, 664, 842, 698, 846, 694, 850, 632, 856, 684, 858, 682, 860, 683, 859, 681, 863, 680, 862, 686, 861, 685, 855, 689, 851, 693, 911, 665, 881, 1003, 647, 953, 651, 893, 2000, 667, 933, 671, 929, 672, 936, 673, 927, 674, 926, 691, 853, 1001, 657, 943, 679, 921, 696, 852, 688, 866, 1000, 654, 886, 658, 882, 1006, 670, 952, 714, 930, 692, 914, 697, 912, 699, 915, 745, 919, 734, 902, 726, 940, 690, 932, 750, 906, 736, 909, 727, 905, 751, 917, 740, 910, 743, 913, 771, 916, 749, 922, 706, 920, 746, 924, 752, 970, 716, 934, 719, 931, 753, 935, 757, 923, 759, 937, 755, 925, 675, 941, 763, 939, 761, 944, 760, 942, 764, 946, 770, 947, 769, 991, 773, 949, 762, 999, 765, 995, 2225, 975, 791, 969, 792, 974, 790, 972, 754, 1010, 756, 1012, 774, 950, 772, 954, 2248, 778, 1066, 780, 1064, 781, 1063, 785, 1061, 783, 1065, 883, 1005, 847, 1004, 867, 1002, 776, 990, 779, 957, 775, 951, 3133, 868, 1011, 837, 1007, 857, 1015, 871, 1008, 870, 1016, 872, 1017, 873, 1018, 880, 1013, 875, 1023, 865, 1021, 877, 1022, 876, 1032, 878, 1020, 887, 1024, 888, 1026, 884, 1027, 5649, 795, 965, 797, 963, 799, 961, 3119, 885, 1041, 817, 1031, 874, 1030, 3414, 687, 1113, 768, 1076, 788, 1056, 3388, 896, 2006, 782, 1084, 786, 1058, 3386, 869, 2002, 784, 1060, 787, 1057, 807, 1037, 3407,...

Maximilian Hasler was also quick to react:

> I agree with Leo. The first 100 terms are:
 1, 3, 5, 7, 2, 4, 6, 8, 10, 34, 9, 20, 42, 18, 30, 14, 31, 13, 35, 17, 33, 11, 37, 15, 39, 16, 38, 50, 22, 40, 26, 41, 19, 36, 52, 24, 46, 21, 45, 27, 44, 28, 58, 70, 23, 57, 25, 47, 29, 51, 73, 43, 60, 82, 12, 48, 62, 32, 56, 84, 49, 61, 83, 55, 71, 53, 75, 91, 63, 80, 54, 72, 90, 64, 81, 59, 77, 92, 69, 85, 101, 65, 89, 66, 88, 100, 67, 93, 76, 94, 79, 97, 74, 96, 200, 68, 86, 102, 87, 103, as given by the following PARI program: {U=a=0; for(n=0,99, da=Set(digits(a)); for(k=valuation(U+2,2),oo, bittest(U,k) || #setintersect(setunion(da, Set(digits(k))), Set( digits( (a+k)*if( bittest(a+k,0), 5, 1/2) ))) || [a=k; break] ); U+=1<<a; print1(a", "))} Maximilian

(and in private):

> voir: https://oeis.org/A357043 (je n’ai pas pu m’empêcher de donner la version qui commence avec 0, mais

 on peut simplement ignorer ce a(0) = 0 pour obtenir ta version "positive".)

Many thanks, Leo and Maximilian (and Hans, see the image below)

Leo Broukhis on Math-Fun:

> For small bases, the smallest integer never to appear is fairly easy to find, because the sequence settles into a pattern relatively quickly, like 1 3 13 39 121 for base 3, or for base 5, after a few hundred initial terms, into 19414 97772 488164 2441522 12206914 etc.That reminds me of turmites. Leo

  

P.-S. I like the hereunder graph — many thanks again to Leo Broukhis & Michael Branicki)

Cet autoportrait ironique et souriant de Gérard Garouste en couverture d’artpress (septembre 2022) renvoie (aussi) au célèbre paradoxe du menteur : oui, semble dire le personnage, je suis Pinocchio, au nez qui s’allonge quand il ment – or mon propre nez s’allonge, donc... Le taulier a toujours aimé le travail et les expos de Garouste. Qu’il soit remercié à travers une simple suite mathématique. Voici ce que dit Garouste dans le magazine artpress à propos de l’illustration qui ouvre cette page :

« Dans tous les thèmes que j’ai par la suite choisis en peinture, Don Quichotte, ou le Faust de Goethe, il y a toujours le l’occulte, de l’intuition et la lumière de la raison classique. D’où le double tableau du clown blanc et de l’Auguste, Lumière : L’Auguste : Yei Or (2019), qui résume parfaitement cette dichotomie. Je suis parti d’une photographie Harcourt de ces deux clowns. Le clown blanc tient une page où figurent les équations de Maxwell sur la lumière. Sur le chapeau melon de l’Auguste est écrit en hébreu : « que la lumière soit et la lumière fut ». Cela m’amusait. Soit vous comprenez le langage des mathématiques, ce qui n’est pas mon cas, soit vous comprenez l’hébreu. Certains connaissent à la fois les mathématiques et l’hébreu, d’autres aucun des deux, alors il leur reste tout de même le diptyque à regarder. »