Add A to B
the result must be
written with some digits of {A} U {B}
[This message was posted on SeqFans on December
14th, 2010. The wrong terms have been corrected today 15th,
thanks to Jean-Marc Falcoz]
Hello SeqFans,
[Write
the sum of a+b with the digits involved in a+b]
Here is my try (always use the smallest integer
not used so far in the seq):
S=1,10,2,19,72,100,3,20,4,30,5,40,6,50,7,60,8,70,9,12,79,18,...
We see that the result of 1+10 uses only digits
of the set {1,1,0}.
The same with 10+2, which uses some elements of
{1,0,2}.
Again, 2+19 uses elements of {2,1,9} for its result.
And I guess 72 is the smallest integer
respecting the constraint (we
see that 19+72 is 91 which uses for its
transcription only a few
elements of the set {1,9,7,2}
Could someone check the existing terms, please,
and extend (if of interest)?
Best,
É.
-----
P.-S.
The set {0,1,2,3} does
not permit to write 110, for
instance.
__________
Nous
cherchons donc la première suite de nombres entiers tous différents dont la
somme de deux termes successifs A et B s’écrive avec les chiffres produits par
l’union des chiffres de A et B.
Explication.
On
commence avec a(1) = 1 ; quel est le plus petit terme ‘t’ qui, ajouté à 1,
donne un résultat qu’on puisse écrire avec le chiffre ‘1’ et les chiffres de
‘t’ ? C’est 10 –-> en effet 1+10=11 et ‘11’ est écrit en prélevant des
chiffres dans l’ensemble {1,1,0} :
S
= 1, 10, ...
Le
terme suivant est 2 car 12 (somme de ‘2’ et ‘10’) s’écrit avec des chiffres de
l’ensemble {1,0,2} :
S = 1, 10, 2, ...
Le
terme suivant est ‘19’ (il n’y a pas plus petit) car 2+19=21 et 21 s’écrit avec
{2,1,9} :
S = 1, 10, 2, 19, ...
Le
terme suivant est ‘72’ (il n’y a pas plus petit) car 19+72=91 et 91 s’écrit
avec {1,9,7,2} :
S = 1, 10, 2, 19, 72, ...
Nous
voulons que tous les termes de S soient différents les uns des autres : la suite S sera-t-elle, in fine, une permutation des nombres
naturels ?
Jean-Marc Falcoz a calculé les 10000 premiers termes de S (les mille premiers
sont ci-dessous, l’ensemble est tout en bas) – il joint un beau diagramme en
forme de cirrus.
à+
É.
__________
S
= 1, 10, 2, 19, 72, 100, 3, 20, 4, 30, 5, 40, 6, 50, 7, 60, 8, 70, 9, 12, 79, 18,
13, 68, 113, 198, 21, 91, 32, 181, 130, 170, 131, 82, 120, 80, 101, 89, 109,
23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37, 126,
93, 41, 273, 52, 163, 29, 63, 107, 123, 96, 71, 230, 73, 241, 83, 152, 64, 182,
31, 92, 119, 78, 102, 99, 112, 179, 132, 81, 97, 122, 169, 127, 25, 47, 125,
94, 51, 300, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34, 106, 234, 95, 61,
340, 62, 153, 128, 57, 114, 297, 104, 39, 56, 139, 172, 42, 175, 136, 180, 22,
190, 710, 191, 628, 157, 264, 140, 260, 141, 173, 138, 193, 108, 213, 168, 143,
171, 240, 160, 242, 186, 345, 49, 105, 346, 185, 133, 178, 203, 67, 103, 207,
134, 177, 244, 184, 228, 194, 208, 214, 197, 204, 137, 174, 238, 144, 268, 148,
233, 159, 232, 470, 231, 87, 192, 527, 188, 224, 195, 306, 145, 267, 149, 142,
75, 283, 245, 167, 129, 162, 53, 272, 250, 251, 164, 248, 183, 135, 176, 355,
150, 350, 151, 274, 147, 265, 257, 269, 253, 74, 362, 154, 258, 270, 33, 290,
610, 291, 528, 187, 284, 364, 84, 351, 165, 247, 205, 315, 146, 266, 256, 271,
43, 281, 437, 206, 254, 158, 223, 469, 225, 305, 196, 243, 85, 400, 44, 360,
155, 358, 375, 156, 239, 353, 372, 330, 370, 261, 255, 294, 118, 263, 259, 333,
296, 307, 326, 304, 189, 354, 217, 295, 227, 396, 216, 246, 166, 365, 218, 406,
215, 308, 275, 249, 475, 277, 430, 276, 341, 76, 293, 336, 287, 235, 288, 457,
285, 237, 335, 316, 317, 356, 279, 445, 389, 420, 380, 421, 398, 405, 327, 403,
337, 286, 342, 86, 500, 55, 450, 98, 121, 680, 1000, 65, 394, 509, 381, 357,
376, 359, 374, 369, 323, 407, 325, 417, 314, 427, 303, 367, 366, 397, 226, 415,
379, 412, 408, 395, 418, 402, 378, 456, 387, 331, 482, 320, 480, 161, 352, 371,
343, 590, 66, 540, 280, 520, 282, 531, 298, 504, 278, 446, 388, 455, 489, 401,
539, 404, 1010, 201, 689, 309, 581, 220, 580, 221, 598, 310, 591, 321, 498,
385, 373, 363, 318, 495, 328, 491, 348, 486, 377, 386, 252, 361, 262, 584, 399,
506, 439, 501, 409, 481, 332, 593, 910, 199, 702, 219, 572, 813, 210, 690, 77,
600, 88, 700, 1001, 110, 790, 111, 900, 1002, 319, 472, 812, 311, 706, 301,
589, 349, 485, 368, 313, 518, 302, 478, 419, 490, 410, 492, 431, 382, 441, 493,
451, 383, 452, 816, 312, 479, 465, 229, 463, 601, 289, 534, 611, 543, 390, 510,
391, 428, 413, 529, 416, 347, 392, 487, 497, 447, 1023, 209, 681, 905, 621,
541, 393, 532, 721, 524, 713, 461, 603, 299, 607, 329, 467, 1003, 429, 503,
1020, 202, 578, 1009, 212, 579, 1008, 440, 1004, 339, 594, 891, 695, 901, 211,
709, 961, 433, 916, 453, 941, 548, 911, 438, 426, 519, 422, 1058, 222, 1580,
521, 631, 292, 587, 991, 432, 711, 423, 811, 645, 511, 634, 612, 513, 338, 496,
448, 1016, 324, 508, 322, 701, 906, 731, 522, 1086, 344, 599, 334, 609, 951,
443, 1027, 414, 630, 411, 605, 471, 903, 1029, 561, 904, 551, 614, 512, 613,
623, 641, 515, 742, 715, 542, 712, 523, 714, 562, 1018, 424, 1017, 384, 459,
1006, 434, 915, 483, 1025, 425, 499, 435, 914, 545, 1005, 449, 505, 1030, 502,
1019, 571, 924, 918, 650, 912, 547, 831, 514, 651, 814, 573, 1062, 458, 931,
564, 1039, 454, 1028, 552, 1068, 535, 1015, 436, 913, 546, 919, 657, 1013, 468,
1012, 568, 1035, 516, 940, 517, 942, 1038, 442, 1160, 444, 1602, 462, 1140,
460, 1007, 530, 1170, 533, 1037, 526, 741, 633, 1070, 507, 1040, 560, 1041,
549, 1045, 538, 1042, 563, 1067, 536, 1024, 558, 1022, 586, 1054, 596, 971,
625, 1047, 525, 1126, 488, 1052, 673, 1033, 537, 821, 647, 1057, 473, 921, 574,
1063, 567, 1036, 559, 1031, 576, 1026, 582, 1046, 615, 841, 627, 1014, 557,
1158, 597, 981, 697, 1032, 648, 1072, 608, 1050, 494, 1055, 800, 1011, 679,
1081, 569, 1021, 669, 1097, 619, 849, 1056, 484, 1164, 477, 1053, 617, 950,
616, 953, 1082, 678, 1090, 565, 1085, 671, 925, 1080, 602, 1060, 544, 1061,
554, 1096, 670, 1034, 606, 1059, 636, 1048, 556, 1049, 705, 1196, 720, 1083,
675, 1075, 683, 1077, 629, 1073, 652, 917, 659, 1137, 476, 1138, 570, 1130,
575, 1176, 585, 1065, 738, 1069, 637, 1066, 674, 1095, 780, 1091, 718, 960,
716, 851, 727, 1043, 687, 1093, 566, 1074, 632, 1071, 635, 1127, 588, 1197,
682, 1078, 624, 1117, 644, 1182, 466, 1148, 666, 1185, 620, 1180, 464, 1150,
2350, 672, 1088, 719, 859, 1051, 639, 1157, 604, 1238, 694, 1208, 577, 1136,
795, 1084, 676, 1064, 595, 1308, 723, 815, 643, 1087, 693, 1076, 626, 1186,
665, 1291, 654, 1107, 653, 1178, 553, 1172, 685, 1120, 684, 1128, 688, 1124,
748, 1092, 677, 1099, 667, 1129, 768, 1102, 708, 1079, 618, 990, 717, 962,
1153, 698, 1103, 728, 1114, 797, 1104, 839, 1142, 779, 1098, 722, 1190, 724,
751, 827, 861, 725, 1192, 729, 1168, 646, 1218, 668, 1146, 1260, 750, 1261,
807, 1169, 745, 1209, 734, 1179, 638, 1175, 583, 1220, 791, 896, 1044, 1227,
699, 1203, 769, 1145, 796, 1105, 2345, 707, 1194, 737, 1280, 730, 1281, 820,
1191, 726, 1195, 732, 1189, 782, 1109, 781, 997, 801, 907, 871, 909, 761, 825,
1200, 802, 1119, 778, 1139, 758, 1113, 798, 1089, 789, 1101, 809, 1112, 799,
1108, 762, 1159, 756, 1219, 752, 1199, 772, 1149, 765, 1250, 474, 1163, 952,
1167, 592, 1287, 691, 895, 1094, 785, 1193, 658, 1173, 920, 1171, 1240, 770,
1241, 840, 1201, 810, 970, 817, ...
Trop
beau !
Merci
encore, Jean-Marc !
__________
P.S. du 16 décembre :
À ma question de savoir si S est ou non une permutation de N (soit les
nombres naturels 1,2,3,4,5,... ), Jean-Marc croit pouvoir répondre par l’affirmative. Il m’envoie du
coup deux magnifiques auto-stéréogrammes
qui affirment que « 0, 1, 10, 2, 19, 72, 100, 3, 20, 4, ... = N ».
Quelle merveille (l’effet de perspective,
notamment) !
Jean-Marc ajoute (au sujet de la
permutation S = N) :
> J’ai
poussé le calcul jusqu’à 10000. Sur ces 10000 termes, le plus petit entier
non utilisé dans la suite est 5550, il serait utilisé si on allait jusqu’à
50000 par exemple (car ça ferait 555550). Avec le même type de raisonnement
pour chaque plus petit entier non encore utilisé, je ne vois pas pourquoi ta
suite ne serait pas une permutation de N.
[Les
10000 termes de S
calculés par Jean-Marc sont ici]
Claudio Meller, de son côté, calcule les débuts de S(2),
S(3), S(4),... S(9) :
> Hi Eric,
Here are the sequences if we start with the
numbers from 2 to 9;
Best,
Claudio.
S(2) = 2,
10, 1, 20, 3, 29, 63, 18, 13, 68, 100, 4, 30, 5, 40, 6, 50, 7, 60, 8, 70, 9,
12, 79, 108, 72, 19, 78, 102, 80, 101, 89, 109, 23, 69, 27, 14, 28, 54, 17, 24,
58, 117, 124, 90, 11, 200, 15, 36, 26, 37, 126, 93, 41, 273, 52, 163, 148, 16,
35, 48, 116, 45, 38, 46, 115, 236, 59, 34, 106, 234, 95, 61, 300, 21, 91, 32,
181, 130, 170, 131, 82, 120, 81, 97, 122, 99, 112, 179, 92, 31, 182, 42, 172,
85, 243, 71, 96, 123, 107, 203, ...
S(3) = 3,
10, 1, 20, 2, 19, 72, 100, 4, 30, 5, 40, 6, 50, 7, 60, 8, 70, 9, 12, 79, 18,
13, 68, 113, 198, 21, 91, 32, 181, 130, 170, 131, 82, 120, 80, 101, 89, 109,
23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37, 126,
93, 41, 273, 52, 163, 29, 63, 107, 123, 96, 71, 230, 73, 241, 83, 152, 64, 182,
31, 92, 119, 78, 102, 99, 112, 179, 132, 81, 97, 122, 169, 127, 25, 47, 125,
94, 51, 300, 16, 35, 48, 116, 45, ...
S(4) = 4,
10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32, 181,
130, 5, 30, 6, 40, 7, 50, 8, 60, 9, 12, 79, 108, 70, 101, 80, 102, 78, 109, 23,
69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37, 126, 93,
41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34, 106, 234,
95, 61, 300, 22, 180, 25, 47, 125, 94, 51, 364, 62, 153, 82, 120, 81, 97, 122,
99, 112, 179, 92, 31, 182, ...
S(5)
= 5, 10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32,
181, 130, 4, 30, 6, 40, 7, 50, 8, 60, 9, 12, 79, 108, 70, 101, 80, 102, 78,
109, 23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37,
126, 93, 41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34,
106, 234, 95, 61, 300, 22, 180, 25, 47, 125, 94, 51, 364, 62, 153, 82, 120, 81,
97, 122, 99, 112, 179, 92, 31, 182, ...
S(6)
= 6, 10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32,
181, 130, 4, 30, 5, 40, 7, 50, 8, 60, 9, 12, 79, 108, 70, 101, 80, 102, 78,
109, 23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37,
126, 93, 41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34,
106, 234, 95, 61, 300, 22, 180, 25, 47, 125, 94, 51, 364, 62, 153, 82, 120, 81,
97, 122, 99, 112, 179, 92, 31, 182, ...
S(7)
= 7, 10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32,
181, 130, 4, 30, 5, 40, 6, 50, 8, 60, 9, 12, 79, 108, 70, 101, 80, 102, 78,
109, 23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37,
126, 93, 41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34,
106, 234, 95, 61, 300, 22, 180, 25, 47, 125, 94, 51, 364, 62, 153, 82, 120, 81,
97, 122, 99, 112, 179, 92, 31, 182, ...
S(8)
= 8, 10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32,
181, 130, 4, 30, 5, 40, 6, 50, 7, 60, 9, 12, 79, 108, 70, 101, 80, 102, 78,
109, 23, 69, 27, 14, 28, 54, 17, 24, 58, 117, 124, 90, 11, 200, 15, 36, 26, 37,
126, 93, 41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34,
106, 234, 95, 61, 300, 22, 180, 25, 47, 125, 94, 51, 364, 62, 153, 82, 120, 81,
97, 122, 99, 112, 179, 92, 31, 182, ...
S(9)
= 9, 10, 1, 20, 2, 19, 72, 100, 3, 29, 63, 18, 13, 68, 113, 198, 21, 91, 32,
181, 130, 4, 30, 5, 40, 6, 50, 7, 60, 8, 70, 101, 80, 102, 78, 109, 12, 79,
108, 90, 11, 200, 14, 27, 25, 47, 125, 94, 51, 300, 15, 36, 26, 37, 126, 93,
41, 273, 52, 163, 148, 16, 35, 48, 116, 45, 38, 46, 115, 236, 59, 34, 106, 54,
17, 24, 58, 117, 124, 95, 61, 340, 62, 153, 82, 120, 81, 97, 122, 99, 112, 179,
92, 31, 182, 42, 172, 85, 243, 71, 96, ...
Claudio a même calculé la suite résultant d’une
contrainte voisine : ici ce n’est pas la somme A+B qui doit s’écrire avec
les chiffres de A U B, mais le produit AB :
> Hi Eric,
> This is a
similar sequence but instead of sum we take the product:
P = 1, 2, 10, 3, 51, 30, 100,
4, 16, 40, 160, 352, 151, 34, 106, 25, 13, 24, 26, 240, 130, 250, 133, 295,
313, 1000, 5, 19, 50, 102, 6, 21, 60, 127, 171, 241, 175, 109, 45, 181, 450,
187, 400, 166, 3052, 1196, 302, 2865, 1441, 298, 31, 1165, 139, 7, 1015, 70,
1043, 700, 1168, 412, 1702, 125, 57, 834, 570, 1250, 137, 55, 1037, 190, 365,
1900, 367, 1689, 59, 845, 491, 395, 1873, 406, 101, 11, 892, 110, 1001, 1009,
445, 183, 264, 238, 136, 12, 68, 120, 451, 114, 1238, 1035, 9, 351, 90, ...
Jean-Marc a calculé 5000 termes de la suite P de Claudio et fournit encore un diagramme superbe.
> Par
curiosité, j’ai fait tourner mon programme "adapté" (il n’y a qu’un
seul signe à changer en tout et pour tout !)
> Le
plus petit entier non utilisé après 5000 nombres est 78. Je joins le dessin
jusqu’à 5000 :
Merci
à tous!
________
Sommaire des suites sommaires