Visible DigitSums

Hello SeqFans,

... here is another re-ordering of the Naturals:

« The digitsum of a(n) is visible in the smallest a(n+1) not yet present in S »:

S = 1, 10, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 100, 21, 23, 25, 27, 29, 111, 30, 31, 24, 26, 28, 101, 20, 22, 34, 37, 102, 32, 35, 38, 112, 40, 41, 45, 49, 113, 50, 51, 46, 103, ...

... this was computed by hand -- and fun to do!

---

BTW:

The 'equivalent' sequence is much harder to write:

"Digitsum of a(n) is visible in a(n-1)", or (better), "a(n) shows the digitsum of the smallest a(n+1) not yet present in T":

T = 1, 10, 19, 9, 18, 8, 17, 7, 16, 6, 15, 5, 14, 4, 13, 3, 12, 2, 11, 29, 20, 101, 28, 26, 24, 22, 110, 37, 21, ...

More headaches than fun with this one!

Best,

É.

__________

Charles Greathouse quickly answered this (February 22nd, 2010):

110 < 111, so the sequence S is more like:

S = 1, 10, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 100, 21, 23, 25, 27, 29, 110, 20, 22, 24, 26, 28, 101, 32, 35, 38, 111, 30, 31, 34, 37, 102, 33, 36, 39, 112, 40, 41, 45, 49, 113, 50, 51, 46, 103, 42, 56, 114, 60, 61, 47, 115, 57, 120, 43, 67, 130, 44, 48, 121, 54, 59, 140, 52, 70, 71, 58, 131, 53, 68, 141, 62, 78, 150, 63, 69, 151, 72, 79, 116, 80, 81, 89, 117, 90, 91, 104, 55, 105, 64, 106, 73, 107, 82, 108, 92, 118, 109, 210, 83, 119, 211, 74, 311, 65, 411, 66, 122, 75, 123, 76, 132, 86, 142, 77, 143, 84, 124, 87, 152, 85, 133, 97, 160, 127, 310, 94, 134, 88, 161, 98, 170, 128, 511, 137, 611, 138, 125, 148, 135, 93, 126, 95, 144, 96, 153, 99, 180, 129, 212, 145, 410, 154, 510, 136, 610, 147, 312, 146, 711, 139, 213, 156, 412, 157, 313, 167, 149, 214, 171, 159, 155, 811, 710, 158, 314, 168, 215, 178, 162, 169, 163, 810, 179, 172, 910, 1000, 164, 911, 1011, 173, 1100, 182, 1101, 183, 512, 181, 1001, 192, 612, 189, 184, 413, 185, 414, 190, 1002, 193, 513, 191, 1102, 174, 712, 1003, 194, 514, 1004, 165, 812, 1103, 175, 613, 1005, 166, 713, 1104, 176, 614, 1105, 177, 315, 195, 415, 1006, 187, 216, 196, 316, 1007, 186, 515, 1106, ...

... Ooops, indeed, Charles, thank you for the corrected sequence!

... and thanks to Charles, this is now A173821

A nice screen capture (and dynamic plotting page - click here) by Georges Brougnard:

Charles has now computed T too (which is now A173822): « a(n) shows the digitsum of the smallest a(n+1) not yet present in T »

T = 1, 10, 19, 9, 18, 8, 17, 7, 16, 6, 15, 5, 14, 4, 13, 3, 12, 2, 11, 29, 20, 101, 28, 26, 24, 22, 110, 37, 21, 100, 46, 31, 30, 102, 55, 23, 111, 38, 35, 32, 120, 39, 27, 25, 41, 40, 103, 64, 33, 201, 200, 299, 36, 42, 112, 47, 34, 121, 48, 44, 130, 49, 45, 50, 104, 73, 43, 202, ...

And again, here is a screen capture taken from Georges Brougnard’s dynamic plotting page:

__________

Another challenge would be to compute U – an extension of the T-sequence concept:

"Start U with 1, 2. Now a(n) must show the digitsum of [a(n+1) + a(n+2)]. Always use the smallest integer not yet present in U"

As far as I get, U looks like this :

U = 1, 2, 8, 3, 5, 7, 16, 9, ...

Could it be that this is a re-ordering of the Naturals?

__________

... Well yes, it is – and Charles has stricken again! With this beautiful list of the first 10000 terms of U (now A173823 in Neil’s database) - plot of first 100 terms by Georges Brougnard, as before (nice plot of 1000 terms at bottom of page):

U = 1, 2, 8, 3, 5, 7, 16, 9, 6, 12, 21, 18, 82, 17, 27, 25, 36, 14, 10, 30, 34, 68, 35, 45, 57, 46, 4, 11, 20, 54, 47, 56, 50, 55, 49, 64, 26, 77, 24, 19, 84, 33, 29, 73, 28, 15, 38, 31, 13, 87, 52, 63, 41, 61, 39, 66, 42, 72, 40, 70, 51, 91, 22, 23, 78, 32, 48, 53, 59, 43, 65, 37, 67, 44, 62, 140, 60, 89, 115, 92, 76, 86, 74, 58, 75, 95, 108, 99, 71, 100, 105, 85, 69, 83, 79, 122, 94, 80, 127, 88, 104, 102, 106, 111, 93, 98, 103, 112, 96, 113, 109, 101, 107, 110, 116, 119, 90, 81, 126, 125, 97, 124, 117, 120, 121, 134, 130, 135, 139, 144, 148, 118, 133, 146, 128, 129, 131, 142, 114, 143, 123, 152, 157, 137, 165, 154, 132, 153, 147, 138, 155, 145, 149, 161, 163, 158, 172, 151, 136, 166, 162, 141, 171, 167, 164, 176, 173, 186, 182, 150, 191, 175, 185, 174, 195, 200, 160, 229, 156, 159, 180, 168, 201, 184, 205, 199, 190, 170, 181, 178, 209, 177, 183, 194, 193, 204, 210, 179, 220, 192, 197, 208, 189, 215, 198, 203, 202, 187, 211, 219, 188, 217, 196, 207, 213, 266, 206, 214, 265, 224, 276, 223, 278, 221, 281, 218, 222, 237, 262, 240, 261, 438, 461, 249, 252, 270, 230, 272, 327, 274, 228, 283, 216, 287, 242, 260, 439, 460, 251, 169, 232, 225, 374, 305, 296, 706, 295, 306, 234, 267, 332, 268, 733, 256, 246, 254, 247, 253, 258, 541, 259, 241, 263, 436, 463, 539, 464, 238, 362, 336, 264, 738, 285, 226, 277, 233, 269, 330, 273, 729, 282, 231, 290, 309, 294, 318, 303, 227, 775, 235, 366, 323, 679, 280, 320, 291, 368, 244, 257, 442, 357, 644, 358, 645, 359, 646, 356, 647, 355, 245, 255, 444, 445, 558, 446, 354, 649, 353, 250, 752, 236, 275, 324, 286, 413, 288, 292, 310, 212, 239, 304, 299, 703, 298, 312, 300, 243, 759, 271, 339, 361, 341, 659, 343, 377, 625, 381, 418, 382, 284, 417, 293, 308, 313, 289, 345, 365, 637, 367, 333, 669, 342, 279, 722, 302, 386, 615, 387, 321, 379, 317, 383, 363, 248, 352, 347, 654, 348, 655, 370, 634, 372, 630, 371, 631, 369, 633, 375, 627, 384, 316, 385, 297, 314, 307, 301, 399, 601, 401, 599, 404, 397, 606, 396, 609, 393, 408, 402, 398, 603, 405, 597, 406, 395, 608, 394, 611, 391, 322, 380, 389, 411, 390, 350, 451, 551, 449, 555, 448, 556, 447, 557, 455, 549, 454, 550, 453, 560, 443, 561, 441, 559, 462, 542, 459, 544, 468, 535, 470, 532, 471, 530, 473, 529, 474, 426, 577, 412, 592, 311, 319, 331, 435, 565, 437, 567, 457, 547, 456, 548, 465, 335, 668, 334, 376, 626, 388, 511, 491, 420, 480, 458, 545, 467, 536, 469, 533, 472, 568, 433, 571, 431, 569, 434, 466, 537, 475, 527, 476, 523, 482, 315, 485, 340, 663, 349, 653, 364, 638, 373, 328, 674, 337, 666, 484, 521, 486, 489, 514, 493, 421, 479, 415, 588, 416, 591, 325, 575, 422, 582, 403, 598, 414, 590, 351, 360, 640, 392, 613, 407, 326, 677, 346, 660, 450, 564, 440, 563, 737, 483, 519, 488, 494, 509, 499, 505, 498, 506, 497, 507, 496, 508, 495, 512, 492, 429, 572, 481, 520, 487, 478, 525, 490, 610, 425, 580, 423, 581, 378, 622, 430, 528, 502, 538, 562, 553, 452, 552, 578, 432, 574, 427, 576, 524, 500, 501, 503, 510, 593, 515, 516, 543, 400, 602, 410, 595, 477, 531, 583, 419, 585, 517, 526, 409, 589, 428, 579, 424, 584, 518, 504, 566, 546, 594, 607, 513, 600, 619, 683, 596, 604, 616, 587, 329, 675, 540, 573, 628, 522, 618, 692, 586, 554, 570, 632, 671, 730, 672, 658, 662, 639, 681, 534, 617, 684, 636, 605, 697, 344, 661, 1340, 664, 338, 665, 648, 753, 650, 651, 749, 652, 678, 623, 680, 721, 682, 621, 620, 691, 629, 673, 641, 670, 731, 689, 614, 688, 612, 690, 711, 693, 709, 695, 635, 667, 643, 687, 624, 696, 705, 699, 701, 700, 702, 718, 783, 719, 782, 686, 716, 685, 708, 704, 698, 713, 717, 694, 726, 714, 786, 715, 785, 724, 676, 735, 765, 736, 764, 656, 745, 755, 746, 854, 1148, 855, 707, 723, 778, 642, 760, 740, 761, 839, 1165, 836, 710, 720, 712, 757, 744, 766, 834, 1170, 831, 725, 777, 823, 1182, 767, 657, 754, 747, 763, 837, 1164, 841, 732, 770, 830, 1175, 826, 739, 772, 758, 743, 768, 742, 769, 741, 762, 748, 852, 848, 1152, 850, 776, 825, 1179, 790, 751, 750, 1253, 779, 727, 774, 1125, 877, 728, 773, 827, 1174, 815, 789, 756, 844, 1159, 843, 788, 813, 787, 814, 796, 784, 816, 794, 734, 866, 1135, 869, 771, 840, 1183, 819, 791, 803, 797, 804, 798, 812, 799, 802, 800, 801, 809, 792, 810, 891, 811, 889, 832, 868, 1132, 872, 833, 867, 1134, 870, 781, 820, 780, 806, 795, 905, 805, 896, 1107, 897, 824, 876, 822, 878, 818, 882, 846, 1154, 857, 859, 1144, 862, 808, 793, 817, 884, 807, 893, 1112, 894, 828, 873, 1126, 875, 829, 871, 929, 1076, 924, 886, 902, 1102, 898, 903, 1103, 904, 906, 1096, 908, 892, 1114, 901, 910, 890, 821, 879, 888, 912, 1094, 913, 887, 845, 1160, 842, 861, 1139, 865, 838, 1166, 835, 895, 1106, 900, 911, 1105, 920, 881, 858, 1157, 849, 863, 847, 853, 1149, 951, 851, 949, 1054, 948, 952, 1050, 950, 860, 940, 1064, 938, 962, 1039, 961, 939, 1065, 936, 874, 926, 1079, 909, 991, 1016, 1000, 915, 985, 947, 1056, 946, 864, 1138, 922, 880, 969, 1037, 967, 933, 1071, 930, 970, 1031, 974, 1027, 975, 925, 1078, 918, 883, 899, 1111, 914, 987, 917, 1088, 919, 981, 856, 944, 1059, 943, 957, 1044, 959, 941, 1062, 945, 955, 1047, 956, 1046, 958, 942, 1061, 964, 1040, 971, 1049, 965, 935, 1068, 953, 1051, 979, 921, 1084, 916, 885, 931, 1072, 980, 1020, 986, 960, 1053, 954, 1058, 963, 937, 1067, 934, 966, 1035, 972, 928, 1077, 923, 977, 990, 1015, 992, 1011, 989, 1019, 988, 984, 1022, 993, 907, 1100, 932, 968, 1032, 983, 1017, 998, 927, 1080, 1024, 982, 995, 1005, 1002, 1001, 999, 1009, 1007, 1010, 1004, 1013, 1097, 1003, 1014, 994, 1008, 1012, 996, 1021, 1082, 1025, 1026, 1018, 1085, 1029, 997, 1028, 1006, 1036, 1089, 973, 1033, 1073, 1034, 976, 1041, 1063, 1038, 1042, 1074, 1060, 1083, 1069, 1045, 1070, 1055, 1052, 978, 1030, 1075, 1086, 1048, 1023, 1057, 1043, 1091, 1109, 1115, 1090, 1081, 1124, 1098, 1092, 1113, 1093, 1087, 1118, 1095, 1101, 1104, 1066, 1140, 1120, 1099, 1108, 1116, 1117, 1122, 1121, 1123, 1129, 1133, 1110, 1142, 1119, 1127, 1143, 1128, 1136, 1167, 1141, 1161, 1130, 1131, 1150, 1156, 1163, 1137, 1147, 1155, 1151, 1168, 1169, 1146, 1158, 1178, 1173, 1184, 1176, 1145, 1188, 1172, 1177, 1162, 1153, 1171, 1180, 1186, 1181, 1195, 1190, 1187, 1199, 1202, 1193, 1185, 1192, 1210, 1204, 1200, 1196, 1191, 1209, 1194, 1212, 1211, 1213, 1201, 1205, 1219, 1214, 1189, 1208, 1197, 1218, 1203, 1217, 1198, 1226, 1206, 1227, 1215, 1236, 1224, 1265, 1223, 1228, 1232, 1247, 1242, 1257, 1248, 1252, 1249, 1256, 1243, 1262, 1237, 1268, 1221, 1230, 1239, 1250, 1255, 1259, 1246, 1274, 1225, 1235, 1207, 1244, 1216, 1283, 1222, 1220, 1231, 1229, 1260, 1245, 1269, 1254, 1251, 1263, 1278, 1272, 1233, 1267, 1238, 1276, 1234, 1266, 1275, 1329, 1284, 1240, 1261, 1271, 1270, 1280, 1324, 1289, 1300, 1241, 1292, 1258, 1343, 1279, 1281, 1320, 1293, 1301, 1303, 1302, 1312, 1311, 1294, 1309, 1295, 1306, 1298, 1307, 1297, 1308, 1296, 1264, 1277, 1273, 1331, 1282, 1323, 1290, 1315, 1286, 1285, 1319, 1321, 1317, 1288, 1326, 1287, 1318, 1304, 1310, 1313, 1328, 1322, 1382, 1349, 1355, 1345, 1305, 1327, 1314, 1291, 1332, 1299, 1333, 1367, 1337, 1376, 1346, 1358, 1364, 1439, 1373, 1341, 1363, 1350, 1354, 1359, 1372, 1368, 1336, 1377, 1426, 1386, 1403, 1400, 1404, 1409, 1395, 1418, 1385, 1338, 1366, 1347, 1357, 1356, 1348, 1365, 1339, 1374, 1330, 1383, 1420, 1392, 1391, 1412, 1490, 1411, 1393, 1408, 1413, 1427, 1394, 1405, 1335, 1316, 1325, 1379, 1334, 1370, 1352, 1361, 1442, 1388, 1344, 1360, 1353, 1351, 1362, 1342, 1371, 1369, 1434, 1378, 1435, 1467, 1436, 1476, 1445, 1458, 1454, 1449, 1463, 1440, 1472, 1431, 1481, 1422, 1499, 1430, 1401, 1421, 1410, 1419, 1381, 1423, 1380, 1397, 1406, 1415, 1389, 1416, 1387, 1417, 1485, 1448, 1455, 1457, 1446, 1466, 1437, 1475, 1428, 1484, 1505, 1398, 1407, 1396, 1444, 1556, 1473, 1529, 1471, 1433, 1470, 1451, 1452, 1460, 1443, 1469, 1461, 1540, 1390, 1424, 1478, 1399, 1414, 1488, 1513, 1492, 1508, 1493, 1429, 1375, 1438, 1464, 1536, 1465, 1535, 1468, 1532, 1480, 1520, 1494, 1487, 1425, 1496, 1506, 1495, 1507, 1503, 1501, 1510, 1538, 1384, 1547, 1453, 1548, 1462, 1539, 1482, 1518, 1483, 1519, 1402, 1479, 1522, 1491, 1447, 1456, 1474, 1527, 1477, 1502, 1511, 1500, 1517, 1531, 1489, 1550, 1497, 1514, 1515, 1486, 1516, 1498, 1504, 1525, 1523, 1534, 1432, 1568, 1524, 1526, 1441, 1528, 1537, 1564, 1546, 1554, 1512, 1545, 1509, 1530, 1521, 1563, 1566, 1544, 1459, 1542, 1541, 1543, 1557, 1553, 1559, 1551, 1533, 1560, 1569, 1578, 1572, 1450, 1552, 1549, 1561, 1562, 1558, 1571, 1567, 1580, 1576, 1555, 1565, 1573, 1574, 1582, 1583, 1591, 1592, 1600, 1604, 1589, 1577, 1570, 1586, 1579, 1595, 1588, 1612, 1607, 1594, 1581, 1575, 1590, 1584, 1599, 1593, 1611, 1602, 1608, 1585, 1616, 1587, 1597, 1603, 1601, 1609, 1620, 1618, 1629, 1627, 1613, 1598, 1610, 1621, 1617, 1630, 1626, 1639, 1635, 1596, 1605, 1614, 1615, 1623, 1606, 1632, 1624, 1641, 1633, 1650, 1642, 1658, 1625, 1622, 1634, 1631, 1643, 1640, 1652, 1648, 1644, 1656, 1636, 1638, 1645, 1647, 1653, 1657, 1662, 1666, 1671, 1675, 1680, 1684, 1689, 1693, 1619, 1628, 1637, 1646, 1654, 1665, 1663, 1674, 1672, 1683, 1681, 1692, 1690, 1701, 1708, 1711, 1699, 1651, 1649, 1670, 1660, 1659, 1669, 1661, 1667, 1679, 1676, 1688, 1685, 1697, 1694, 1706, 1696, 1678, 1668, 1687, 1677, 1705, 1686, 1714, 1695, 1712, 1715, 1704, 1713, 1724, 1682, 1710, 1691, 1709, 1655, 1719, 1717, 1702, 1698, 1721, 1733, 1722, 1742, 1727, 1673, 1746, 1726, 1720, 1735, 1723, 1732, 1741, 1750, 1759, 1768, 1777, 1786, 1795, 1813, 1787, 1664, 1728, 1744, 1729, 1753, 1738, 1771, 1747, 1762, 1756, 1780, 1765, 1789, 1774, 1736, 1737, 1700, 1755, 1718, 1764, 1745, 1773, 1754, 1782, 1763, 1791, 1772, 1800, 1781, 1783, 1798, 1792, 1808, 1809, 1818, 1827, 1836, 1845, 1863, 1854, 1872, 1881, 1890, 1917, 1899, 1810, 1790, 1730, 1707, 1739, 1716, 1703, 1725, 1748, 1734, 1757, 1752, 1766, 1743, 1775, 1761, 1784, 1770, 1793, 1779, 1731, 1751, 1740, 1769, 1749, 1760, 1758, 1778, 1767, 1796, 1776, 1805, 1785, 1788, 1802, 1806, 1803, 1815, 1794, 1797, 1811, 1824, 1812, 1833, 1821, 1842, 1830, 1851, 1839, 1869, 1848, 1860, 1857, 1878, 1866, 1887, 1875, 1896, 1884, 1923, 1893, 1816, 1820, 1807, 1822, 1814, 1804, 1801, 1817, 1819, 1799, 1828, 1825, 1829, 1834, 1838, 1843, 1847, 1861, 1856, 1852, 1865, 1870, 1874, 1879, 1883, 1888, 1892, 1915, 1901, 1826, 1910, 1897, 1911, 1905, 1831, 1914, 1902, 1906, 1920, 1924, 1840, 1919, 1933, 1929, 1942, 1938, 1969, 1947, 1960, 1956, 1951, 1965, 1978, 1974, 1987, 1983, 2020, 1932, 1823, 1832, 1849, 1841, 1867, 1850, 1858, 1859, 1876, 1868, 1885, 1877, 1894, 1886, 1921, 1895, 1913, 1903, 1908, 1837, 1926, 1935, 1844, 1928, 1996, 1992, 2010, 1995, 1922, 1904, 1835, 1937, 2005, 1998, 1946, 1862, 1955, 2047, 1954, 1855, 1953, 1962, 1846, 1944, 1971, 1864, 1980, 2022, 1912, 2007, 1873, 1882, 1853, 1891, 1871, 1900, 1880, 1964, 2040, 1930, 2014, 1985, 1941, 1939, 1968, 1948, 1959, 1957, 1950, 1966, 1977, 1975, 1986, 1984, 2018, 1889, 1918, 1898, 1909, 1907, 1973, 1934, 1982, 1925, 1991, 1916, 2000, 1931, 2004, 2001, 1943, 2019, 1981, 1927, 1989, 1936, 2016, 1945, 1963, 2037, 2013, 1940, 1961, 2009, 1952, 2028, 2041, 1972, 1990, 2015, 1999, 2002, 2003, 2076, 2012, 2008, 2027, 2052, 2036, 2043, 2045, 2034, 1967, 2033, 1970, 2109, 1993, 1979, 2026, 1988, 2023, 2017, 2062, 2032, 2056, 2113, 1997, 2021, 1949, 2031, 2038, 2050, 2029, 2059, 2054, 2025, 1976, 2103, 2065, 2024, 2055, 2042, 2046, 2051, 2118, 2060, 2035, 2044, 2053, 2116, 2071, 2030, 2049, 2039, 2011, 1994, 2058, 2064, 1958, 2121, 2074, 2006, 2073, 2096, 2082, 2087, 2091, 2078, 2100, 2069, 2063, 2068, 2101, 2077, 2102, 2067, 2083, 2086, 2092, 2095, 2108, 2061, 2080, 2089, 2098, 2104, 2099, 2105, 2107, 2072, 2048, 2130, 2057, 2114, 2145, 2081, 2127, 2075, 2122, 2079, 2110, 2093, 2111, 2085, 2066, 2112, 2084, 2131, 2128, 2088, 2123, 2097, 2136, 2094, 2106, 2090, 2132, 2154, 2115, 2139, 2070, 2133, 2126, 2117, 2125, 2140, 2120, 2124, 2135, 2134, 2144, 2143, 2119, 2150, 2137, 2141, 2129, 2142, 2160, 2147, 2149, 2138, 2158, 2148, 2153, 2157, 2151, 2166, 2169, 2167, 2178, 2176, 2152, 2185, 2159, 2170, 2156, 2163, 2172, 2146, 2173, 2161, 2194, 2165, 2155, 2181, 2187, 2190, 2175, 2164, 2182, 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Image of the first 1000 terms of U (coming from here:)

Bravo and thank you, Charles and Georges, great job!

Best,

É.

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