[seqfan] Re: smallest number which is coprime to the last three predecessors and not occurring earlier;

Tue Jan 18 17:28:34 CET 2011

> 1,2,3,5,7,4,9,10,11,...
>
> is that the right start for smallest number which is coprime to the
> last three predecessors and not occurring earlier; a(1)=1, a(2)=2,
> a(3) = 3?

Yes.  A few more terms:

1, 2, 3, 5, 7, 4, 9, 11, 13, 8, 15, 17, 19, 14, 23, 25, 27, 16, 29,
31, 21, 10, 37, 41, 33, 20, 43, 47, 39, 22, 35, 53, 51, 26, 49, 55,
57, 32, 59, 61, 45, 28, 67, 71, 65, 6, 73, 77, 79, 12, 83, 85, 89, 18,
91, 95, 97, 24, 101, 103, 107, 30, 109, 113, 119, 36, 115, 121, 127,
34, 63, 125, 131, 38, 69, 137, 139, 40, 81, 133, 143, 46, 75, 149,
151, 44, 87, 155, 157, 52, 99, 145, 161, 62, 111, 163, 167, 50, 93,
169, 173, 56, 123, 179, 181, 58, 105, 187, 191, 64, 117, 175, 193, 68,
129, 185, 197, 76, 141, 199, 203, 74, 135, 209, 211, 82, 147, 215,
221, 88, 159, 205, 217, 86, 153, 223, 227, 70, 171, 229, 233, 80, 177,
239, 241, 92, 165, 247, 251, 94, 183, 245, 253, 104, 201, 235, 257,
98, 207, 263, 265, 112, 213, 269, 271, 100, 189, 277, 281, 106, 195,
259, 283, 116, 219, 275, 287, 118, 237, 289, 293, 110, 243, 299, 301,
122, 225, 307, 311, 124, 231, 295, 313, 128, 249, 305, 317, 134, 261,
323, 325, 142, 267, 319, 329, 130, 279, 331, 337, 136, 273, 335, 341,
146, 291, 343, 347, 148, 255, 349, 353, 152, 297, 355, 359, 158, 303,
361, 365, 154, 309, 367, 373, 140, 321, 377, 379, 160, 327, 371, 383,
164, 285, 389, 391, 166, 315, 397, 401, 172, 333, 385, 403, 178, 339,
395, 407, 182, 369, 409, 415, 176, 351, 413, 419, 170, 363, 421, 427,
184, 375, 431, 433, 188, 345, 439, 443, 194, 357, 437, 445, 202, 381,
449, 425, 196, 387, 451, 457, 190, 393, 461, 463, 200, 399, 467, 473,
206, 405, 469, 479, 208, 411, 475, 487, 212, 417, 455, 491, 214, 423,
481, 485, 218, 441, 493, 499, 220, 447, 497, 503, 226, 429, 505, 509,
224, 453, 515, 517, 232, 459, 511, 521, 230, 471, 523, 527, 236, 435,
529, 533, 238, 465, 541, 547, 242, 477, 535, 551, 244, 483, 545, 557,
248, 489, 539, 559, 250, 501, 553, 563, 254, 495, 569, 571, 256, 507,
565, 577, 262, 513, 575, 581, 268, 519, 583, 587, 260, 531, 589, 593,
272, 525, 599, 601, 274, 537, 595, 607, 278, 543, 605, 611, 266, 549,
613, 617, 280, 561, 619, 631, 284, 555, 623, 641, 286, 573, 625, 629,
292, 567, 635, 643, 296, 579, 637, 647, 290, 591, 649, 653, 298, 585,
659, 661, 302, 597, 655, 667, 304, 603, 671, 673, 310, 609, 677, 683,
314, 615, 679, 689, 316, 621, 665, 691, 326, 633, 685, 697, 308, 639,
695, 701, 322, 627, 709, 719, 320, 651, 703, 727, 328, 645, 707, 713,
332, 657, 715, 721, 334, 669, 725, 731, 338, 681, 733, 737, 340, 687,
739, 743, 344, 663, 745, 749, 346, 699, 751, 755, 352, 711, 757, 761,
350, 717, 767, 769, 356, 675, 763, 773, 358, 705, 779, 781, 362, 723,
775, 787, 364, 729, 785, 797, 368, 693, 793, 799, 370, 747, 791, 803,
376, 741, 805, 809, 374, 753, 811, 815, 382, 759, 817, 821, 386, 735,
823, 827, 388, 765, 829, 839, 392, 771, 835, 841, 394, 777, 845, 853,
398, 783, 833, 851, 380, 789, 847, 857, 400, 801, 859, 863, 404, 795,
869, 871, 406, 807, 865, 877, 412, 813, 875, 881, 416, 831, 883, 887,
410, 819, 893, 899, 422, 825, 889, 901, 428, 837, 895, 907, 418, 843,
905, 911, 424, 849, 913, 917, 430, 867, 919, 923, 434, 855, 929, 937,
436, 861, 925, 941, 442, 873, 931, 943, 440, 879, 947, 949, 446, 885,
953, 959, 452, 891, 955, 961, 448, 897, 935, 967, 454, 903, 965, 971,
458, 909, 973, 977, 460, 921, 979, 983, 464, 915, 989, 991, 466, 927,
985, 997, 472, 933, 995, 1001, 478, 939, 1003, 1007, 470, 951, 1009,
1013, 476, 957, 1019, 1021, 482, 945, 1027, 1031, 484, 963, 1015,
1033, 488, 969, 1025, 1039, 496, 981, 1037, 1043, 494, 993, 1049,
1051, 490, 999, 1061, 1063, 500, 987, 1067, 1069, 502, 975, 1057,
1073, 506, 1005, 1079, 1087, 508, 1011, 1045, 1081, 512, 1017, 1055,
1091, 514, 1023, 1075, 1093, 518, 1041, 1097, 1103, 520, 1029, 1109,
1111, 524, 1035, 1099, 1117, 526, 1047, 1085, 1121, 536, 1053, 1115,
1123, 532, 1059, 1105, 1129, 538, 1077, 1127, 1133, 530, 1083, 1139,
1141, 542, 1065, 1147, 1151, 544, 1089, 1135, 1153, 548, 1071, 1145,
1157, 554, 1101, 1159, 1163, 550, 1107, 1169, 1171, 556, 1095, 1177,
1181, 562, 1113, 1165, 1187, 566, 1119, 1175, 1183, 568, 1137, 1189,
1193, 560, 1143, 1199, 1201, 574, 1125, 1207, 1213, 572, 1149, 1195,
1211, 578, 1131, 1205, 1217, 584, 1161, 1219, 1223, 580, 1167, 1229,
1231, 586, 1155, 1237, 1241, 592, 1179, 1225, 1243, 596, 1173, 1235,
1247, 604, 1191, 1249, 1253, 590, 1203, 1259, 1261, 602, 1185, 1271,
1273, 598, 1215, 1267, 1277, 608, 1209, 1255, 1279, 614, 1197, 1265,
1283, 622, 1227, 1285, 1289, 616, 1233, 1291, 1297, 610, 1221, 1301,
1303, 620, 1239, 1307, 1313, 626, 1245, 1309, 1319, 628, 1251, 1295,
1321, 632, 1257, 1315, 1327, 634, 1263, 1325, 1331, 644, 1269, 1333,
1339, 638, 1275, 1337, 1349, 652, 1287, 1343, 1345, 656, 1281, 1357,
1355, 646, 1293, 1351, 1361, 640, 1299, 1363, 1367, 650, 1311, 1369,
1373, 658, 1305, 1381, 1387, 662, 1317, 1375, 1379, 664, 1329, 1385,
1391, 668, 1323, 1397, 1399, 670, 1341, 1393, 1403, 674, 1335, 1409,
1411, 676, 1347, 1405, 1421, 682, 1359, 1415, 1417, 686, 1353, 1423,
1427, 680, 1371, 1429, 1433, 688, 1365, 1439, 1441, 692, 1377, 1435,
1447, 694, 1383, 1445, 1451, 698, 1389, 1453, 1457, 700, 1401, 1459,
1469, 704, 1395, 1471, 1477, 706, 1413, 1465, 1463, 712, 1431, 1475,
1481, 716, 1407, 1483, 1487, 710, 1419, 1489, 1493, 718, 1425, 1499,
1507, 724, 1437, 1495, 1501, 734, 1461, 1505, 1511, 722, 1443, 1513,
1519, 730, 1467, 1517

> Is it still true that primes occur in natural order?

Yes.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Jan 17, 2011 at 7:44 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> 1,2,3,5,7,4,9,10,11,...
>
> is that the right start for smallest number which is coprime to the
> last three predecessors and not occurring earlier; a(1)=1, a(2)=2,
> a(3) = 3?
>
> This would be to 3 as A084937 is to 2.
>
> It would be a permutation of the natural numbers.
>
> Is it still true that primes occur in natural order?
>
> Would it also have few fixed points?
>
> -- Jonathan Vos Post
>
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