Keep the integer parts, concatenate them
Hello Math-Fun,
Rules:
a) For the successive pairs {a(n);a(n+1)} of S:
– always divide the large number by the small one,
– keep the integer part of the division,
– form T with those successive integer parts.
b) S and T must share the same digit-succession.
– keep the integer part of the division,
– form T with those successive integer parts.
b) S and T must share the same digit-succession.
S = 1, 11, 6, 36, 12, 2, 3, 7, 14, 4, 27, 15, 60, 13, 5, 40, 21, 105, 6300, 484, 96, 3840, 182, 19110, 3185, 955500, 1974, 17766, 2961, 77, …
Check
pair 1: {1;11} —> 11/1 == 11
pair 2: {11;6} —> 11/6 == 1
pair 3: {6;36} —> 36/6 == 6
pair 4: {36;12} —> 36/12 == 3
pair 5: {12;2} —> 12/2 == 6
pair 6: {2;3} —> 3/2 == 1
pair 7: {3;7} —> 7/3 == 2
pair 8: {7;14} —> 14/7 == 2
pair 9: {14;4} —> 14/4 == 3
pair 10: {4;28} —> 28/4 == 7
…
T is formed by the successive results to the right of the above == sign.
T = 11, 1, 6, 3, 6, 1, 2, 2, 3, 7, …
Comparison with S:
S = 1, 11, 6, 36, 12, 2, 3, 7, …
The digit-successions of S and T are indeed the same.
Question
Is the above S the lexicographically earliest seq of distinct terms > 0 obeying the rules? In verifying this, mind the zeros of S — which must reappear at some point in T (inside or at the end of a term of T).
Best,
É.
P.-S.
S is not in the OEIS — but as I am not sure of my hand calculations, this is perhaps old hat.
A sequence more lexicographic is this one:
RépondreSupprimer[1, 11, 6, 36, 10, 60, 600, 36000, 21600000, 5400001, 900, 41, 24600000, 4100001…]
But, it becomes very difficult to go far with the numerous zeros which are expanding.
A variant might be to omit terms ending by zero. So the sequence is much more fluid:
[1, 11, 6, 36, 12, 2, 3, 7, 14, 4, 28, 15, 61, 13, 5, 41, 21, 105, 16, 9, 8, 24, 121, 25, 17, 34, 18, 181, 31, 19, 114, 1026, 115, 39, 156, 79, 27, 22, 44, 221, 111, 777, 195, 42, 23, 184, 93, 744, 54, 29, 26, 234, 118, 62, 248, 2481, 92, 47, 32, 161, 43, 387, 194, 33, 198, 1386, 139, 48, 336, 113, 38, 152, 35, 71, 142, 72, 37, 45, 46, 322, 2254, 282, 143, 1287, 215, 49, 98, 196, 51, 52, 416, 84, 756, 191, 1337, 268, 55, 275, 56, 112, 1008, 337, 53, 106, 318, 64, 57, 58, 464, 67, 134, 59, 236, 1888, 76, 608, 305, 2745, 916, 185, 1295, 324, 109, 63, 378, 192, 768, 193, 579, 65, 455, 228, 2052, 411, 103, 309, 155, 1395, 157, 81, 243, 1944, 278, 141, 423, 3807, 78, 235, 66, 396, 199, 101, 303, 77, 616, 311, 1555, 519, 131, 655, 82, 68, 69, 276, 94, 658, 222, 73, 511, 104, 521, 107, 642, 162, 74, 148, 296, 99, 495, 102, 204, 1632, 75, 301, 83, 85, 171, 1368, 172, 86, 87, 435, 88, 792, 7128, 793, 397, 3573, 512, 89, 91, 456, 153, 612, 307, 1842, 205, 821, 108, 541, 3246, 163, 95, 96, 288, 864, 116, 232, 1392, 158, 791, 132, 264, 1848, 312, 1561, 224, 117, 97, 197, 19701, 237, 711, 4977, 831, 208, 2081, 298, 894, 448, 3584, 513, 119, 595, 4165, 695, 5561, 1113, 164, 656, 3936, 493, 247, 741, 149, 745, 6705, 281, 1686, 844, 6752, 751, 6008, 752, 122, 7321, 814, 24421, 461, 3227, 646, 123, 1107, 554, 3324, 175, 875, 438, 147, 1323, 223, 669, 225, 901, 9011, 902, 129, 388, 124, 868, 6944, 348, 125, 876, 126, 1009, 505, 4545, 127, 889, 8001, 1144, 201, 804, 135, 675, 226, 452, 3616, 173, 865, 289, 1156, 581, 291, 2911, 728, 21841, 238, 1191, 202, 128, 384, 3456, 577, 292, 1461, 183, 1464, 733, 245, 981, 246, 133, 1197, 241, 964, 323, 2261, 252, 136, 544, 273, 1092, 365, 1095, 274, 21921, 283, 2264, 755, 189, 945, 137, 822, 206, 1854, 265, 138, 1242, 11178, 1017, 509, 15271, 402, 2814, 403, 203, 1218, 306, 154, 144, 145, 725, 3625, 605, 3025, 159, 146, 439, 227, 1362, 229, 1145, 9161, 341, 2728, 391, 3519, 1174, 151, 906, 8154, 174, 871, 6968, 304, 609, 4263, 1066, 178, 165, 166, 1661, 333, 1665, 556, 279, 2791, 349, 2094, 419, 167, 168, 1011, 338, 2366, 474, 239, 956, 7648, 255, 1531, 169, 1521, 308, 2772, 463, 4631, 1544, 30881, 736, 4416, 1105, 369, 2583, 431, 12931, 681, 176, 1408, 242, 177, 1239, 621, 313, 939, 5634, 627, 314, 2198, 734, 5872, 839, 6712, 841, 179, 537, 2685, 299, 2392, 302, 2718, 907, 6349, 489, 3912, 491, 4419, 1106, 277, 2493, 315, 946, 182, 1274, 319, 1595, 798, 267, 2136, 214, 1926, 965, 207, 1035, 6211, 389, 1167, 186, 187, 374, 11221, 1403, 702, 5616, 1124, 375, 7501, 1251, 10008, 457, 4571, 508, 2541, 514, 258, 774, 259, 1036, 6216, 366, 1098, 9882, 1648, 14832, 2119, 707, 5656, 629, 5032, 719, 2876, 244, 1465, 492, 1476, 494, 249, 747, 6723, 316, 1581, 188, 1316, 11844, 988, 251, 502, 209, 1254, 253, 211, 1688, 339, 2712, 679, 342, 684, 343, 1715, 254, 212, 424, 848, 3392, 284, 1988, 213, 1491, 746, 6714, 842, 422, 3798, 265861, 2145, 269, 1883, 942, 472, 1889, 17001, 218, 1744, 437, 219, 4381, 487, 9741, 1083, 542, 216, 1945, 217, 1736, 15624, 351, 1404, 11232, 317, 2536, 515, 2575, 1288, 325, 231, 233, 2097, 352, 3168, 529, 2116, 1059, 6354, 1061, 6366, 637, 3185, 531, 2655, 381, 256, 257, 261, 262, 786, 394, 2364, 473, 2838, 475, 2851, 713, 6417, 1605, 263, 1052, 9468, 287, 1148, 8036, 1005, 4021, 344, 1376, 12384, 1549, 321, 1606, 266, 18621, 353, 2824, 1413, 708, 4248, 476, 2856, 326, 1304, 271, 1626, 11382, 1898, 633, 4431, 739, 371, 222601, 2531, 425, 851, 426, 852]