Digit Positions in S

(a self-describing sequence)

Neil Sloane, David Wasserman and Maximilian Hasler have joined their efforts and computer skills to provide you with this bunch of nice self-describing sequences (from the point of view of their digit’s position); many thanks to all of them!

A098645 Start with a(1) = 1. For n>1, choose a(n) to be the smallest number > a(n-1) consistent with the condition that "the a(n)-th digit is a 1 and no 1’s occur in other positions" is true for all n.

S1 = 1, 3, 10, 20, 22, 31, 32, 33, 34, 35, 41, 51, 52, 53, 54, 55, 111, 112, 200, 210, 220, 222, 231, 1111, 2000, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2020, 2022, 2023, 2024, 2031, 10000, 20000, 20002, 20003, 20004, 20005, 20006, 20007, 20008, 20009,...

A167452 Smallest sequence which lists the position of digits "2" in the sequence.

S2 = 3, 4, 22, 30, 31, 33, 34, 35, 36, 37, 38, 42, 43, 44, 45, 52, 202, 222, 223, 302, 2220, 3000, 3200, 3300, 3301, 3303, 3304, 3305, 3306, 3307, 3308, 3309, 3310, 3311, 3313, 3314, 3315, 3316, 3317, 3318, 3319, 3330, 3331, 3333, 3334, 3335, 3336, 3337, 3338,...

A167453 Smallest sequence which lists the position of digits "3" in the sequence.

S3 = 2, 3, 30, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 63, 330, 333, 3333, 33333, 33400, 40300, 40400, 40401, 40402, 40404, 40405, 40406, 40407, 40408, 40409, 40410, 40411, 40412, 40414, 40415, 40416, 40417, 40418, 40419, 40420,...

A167454 Smallest sequence which lists the position of digits "4" in the sequence.

S4 = 2, 4, 5, 44, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 400, 500, 4444, 5444, 44444, 45444, 444000, 500000, 500001, 500002, 500003, 500005, 500006, 500007, 500008, 500009, 500010, 500011, 500012, 500013, 500015, 500016,...

A167455 Smallest sequence which lists the position of digits "5" in the sequence.

S5 = 2, 5, 6, 7, 55, 56, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 550, 605, 5555, 6555, 55555, 56555, 555555, 600000, 600001, 600002, 600003, 600004, 600006, 600007, 600008, 600009, 600010, 600011, 600012, 600013,...

A167456 Smallest sequence which lists the position of digits "6" in the sequence.

S6 = 2, 6, 7, 8, 9, 66, 660, 700, 701, 702, 703, 704, 705, 707, 708, 709, 710, 711, 712, 713, 714, 715, 717, 718, 719, 760, 770, 771, 772, 773, 774, 775, 777, 778, 779, 780, 781, 782, 783, 784, 785, 787, 788, 789, 790, 791, 792, 793, 794, 795, 797, 798, 799, 800,...

A167457 Smallest sequence which lists the position of digits "7" in the sequence.

S7 = 2, 7, 8, 9, 10, 77, 770, 800, 801, 802, 803, 804, 805, 806, 808, 809, 810, 811, 812, 813, 814, 815, 816, 818, 819, 820, 821, 822, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 838, 839, 840, 841, 842, 843, 844, 845, 846, 848, 849, 850, 851, 852, 853, 854,...

A167450 Smallest sequence which lists the position of digits "8" in the sequence.

S8 = 2, 8, 9, 10, 11, 88, 880, 900, 901, 902, 903, 904, 905, 906, 907, 909, 910, 911, 912, 913, 914, 915, 916, 917, 919, 920, 921, 922, 923, 924, 925, 926, 8000, 9000, 9001, 9002, 9003, 9004, 9005, 9006, 9007, 9009, 9010, 9011, 9012, 9013, 9014, 9015, 9016, 9017,...

A167451 Smallest sequence which lists the position of digits "9" in the sequence.

S9 = 2, 9, 10, 11, 12, 99, 990, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1020, 1021, 1022, 1900, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2010, 2011, 2012, 2013, 2014, 2015, 2016,...

A167519 Lexicographically first sequence which lists the positions of the zero digit in the sequence.

S0 = 3, 10, 11, 12, 11000, 11111, 11112, 11113, 11114, 11115, 11116, 11117, 11118, 11119, 11121, 11122, 11123, 11124, 11125, 11126, 11127, 11128, 11129, 11131, 11132, 11133, 11134, 11135, 11136, 11137, 11138, 11139, 11141, 11142, 11143, 11144, 11145, 11146, 11147, ...

__________

As David Wasserman mentions it here A098645, if we drop the constraint “For n>1, choose a(n) to be the smallest number > a(n-1)” and replace it by “For n>1, choose a(n) to be the smallest number not yet used” we’ll get another bunch of such self-describing sequences – but not monotonically increasing.

__________

[end of november 2009 update]:

I’ve posted the above remark on fr.sci.maths group and got this quick reply from “zwim”:

Le Tue, 24 Nov 2009 07:44:48 -0800 (PST) eric.angelini a écrit :

>Hello Fr.Sci.Maths,

>qq'un du forum aurait-il le temps/la patience/le talent

>de calculer les séries évoquées tout en bas de la page, ici :

>Il semblerait que ce soit un casse-tête à programmer...

>Les auteurs seront crédités dans l'OEIS de Neil Sloane, bien sûr.

>à+

>É.

Si je me suis pas trop gouré, voici un programme en C — ce n'est effectivement pas une sinécure à programmer, sans doute est-ce beaucoup plus simple dans un langage fonctionnel...

Et les listes résultats :

Je me suis limité à 1000 digits.

Ça semble coller pour les premiers termes fournis de D=1. J'ai tenté une vérification sommaire à la main pour D=1 et D=2, ça semble correct, mais il faudrait confirmer. Si c'est pas ça, dis-moi ça cloche je corrigerai (si j'y arrive...), c'est un premier jet, donc prudence. C'est assez rigolo, les apparitions de xx, xxx, xxxx, xxxxx, xxxxxx,... à partir d'un certain rang.

Sequence for D=1

1, 3, 10, 6, 11, 7, 21, 13, 15, 17, 19, 101, 24, 100, 29, 102, 34, 103, 39, 104, 44, 105, 49, 106, 54, 107, 59, 108, 64, 109, 69, 110, 70, 76, 111, 77, 78, 85, 112, 86, 91, 94, 113, 95, 211, 1111, 11111, 1110, 115, 116, 118, 119, 121, 122, 124, 125, 127, 129, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 168, 169, 172, 175, 178, 181, 184, 187, 190, 193, 196, 199, 201, 202, 205, 208, 214, 217, 222, 233, 236, 250, 1000, 257, 1001, 260, 267, 1002, 274, 1003, 281, 1004, 280, 291, 1005, 290, 301, 1006, 300, 311, 1007, 309, 310, 319, 322, 330, 1008, 337, 1009, 344, 1010, 346, 354, 1011, 356, 357, 367, 1012, 369, 377, 1013, 379, 387, 1014, 389, 397, 1015, 399, 407, 1016, 409, 415, 418, 423, 1017, 425, 433, 1018, 435, 443, 1019, 445, 453, 1020, 460, 1021, 463, 470, 1022, 477, 1023, 484, 1024, 491, 1025, 490, 501, 1026, 500, 511, 1027, 509, 510, 519, 522, 530, 1028, 537, 1029, 544, 1030, 551, 1031, 550, 554, 564, 1032, 571, 1033, 570, 581, 1034, 580, 591, 1035, 590, 601, 1036, 600, 611, 1037, 609, 610, 619, 622, 630, 1038, 637, 1039, 644, 1040, 651, 1041, 650, 654, 664, 1042, 671, 1043, 670, 681, 1044, 680, 691, 1045, 690, 701, 1046, 700, 711, 1047, 709, 710, 719, 722, 730, 1048, 737, 1049, 744, 1050, 751, 1051, 750, 754, 764, 1052, 771, 1053, 770, 781, 1054, 780, 791, 1055, 790, 801, 1056, 800, 811, 1057, 809, 810, 819, 822, 830, 1058, 837, 1059, 844, 1060, 851, 1061, 850, 854, 864, 1062, 871, 1063, 870, 881, 1064, 880, 891, 1065, 890, 901, 1066, 900, 911, 1067, 909, 910, 919, 922, 930, 1068, 937, 1069, 944, 1070, 951, 1071, 950, 954, 964, 1072, 971, 1073, 970, 981, 1074, 980, 991, 1075, 990, 1211, ...

Mehdi Tibouchi has extended the above sequence to thousand terms (they show 3282 digits “1”):

1, 3, 10, 6, 11, 7, 21, 13, 15, 17, 19, 101, 24, 100, 29, 102, 34, 103, 39, 104, 44, 105, 49, 106, 54, 107, 59, 108, 64, 109, 69, 110, 70, 76, 111, 77, 78, 85, 112, 86, 91, 94, 113, 95, 211, 1111, 11111, 1110, 115, 116, 118, 119, 121, 122, 124, 125, 127, 129, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 168, 169, 172, 175, 178, 181, 184, 187, 190, 193, 196, 199, 201, 202, 205, 208, 214, 217, 222, 233, 236, 250, 1000, 257, 1001, 260, 267, 1002, 274, 1003, 281, 1004, 280, 291, 1005, 290, 301, 1006, 300, 311, 1007, 309, 310, 319, 322, 330, 1008, 337, 1009, 344, 1010, 346, 354, 1011, 356, 357, 367, 1012, 369, 377, 1013, 379, 387, 1014, 389, 397, 1015, 399, 407, 1016, 409, 415, 418, 423, 1017, 425, 433, 1018, 435, 443, 1019, 445, 453, 1020, 460, 1021, 463, 470, 1022, 477, 1023, 484, 1024, 491, 1025, 490, 501, 1026, 500, 511, 1027, 509, 510, 519, 522, 530, 1028, 537, 1029, 544, 1030, 551, 1031, 550, 554, 564, 1032, 571, 1033, 570, 581, 1034, 580, 591, 1035, 590, 601, 1036, 600, 611, 1037, 609, 610, 619, 622, 630, 1038, 637, 1039, 644, 1040, 651, 1041, 650, 654, 664, 1042, 671, 1043, 670, 681, 1044, 680, 691, 1045, 690, 701, 1046, 700, 711, 1047, 709, 710, 719, 722, 730, 1048, 737, 1049, 744, 1050, 751, 1051, 750, 754, 764, 1052, 771, 1053, 770, 781, 1054, 780, 791, 1055, 790, 801, 1056, 800, 811, 1057, 809, 810, 819, 822, 830, 1058, 837, 1059, 844, 1060, 851, 1061, 850, 854, 864, 1062, 871, 1063, 870, 881, 1064, 880, 891, 1065, 890, 901, 1066, 900, 911, 1067, 909, 910, 919, 922, 930, 1068, 937, 1069, 944, 1070, 951, 1071, 950, 954, 964, 1072, 971, 1073, 970, 981, 1074, 980, 991, 1075, 990, 1211, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 111111111110, 998, 1080, 1084, 1088, 1092, 1096, 1100, 1101, 1311, 1104, 1105, 1107, 1108, 1112, 1113, 1116, 1117, 1120, 1121, 1124, 1125, 1128, 1129, 1130, 1132, 1133, 1134, 1136, 1137, 1138, 1140, 1141, 1142, 1151, 1144, 1145, 1148, 1149, 1152, 1153, 1156, 1157, 1160, 1161, 1164, 1165, 1168, 1169, 1172, 1173, 1176, 1177, 1180, 1181, 1184, 1185, 1188, 1189, 1201, 1192, 1193, 1196, 1197, 1200, 1203, 1204, 1205, 1208, 1209, 1212, 1213, 1216, 1217, 1220, 1221, 1224, 1225, 1228, 1229, 1232, 1233, 1236, 1237, 1240, 1241, 1244, 1245, 1248, 1249, 1251, 1252, 1253, 1256, 1257, 1260, 1261, 1264, 1265, 1268, 1269, 1272, 1273, 1276, 1277, 1280, 1281, 1284, 1285, 1288, 1289, 1291, 1292, 1293, 1296, 1297, 1300, 1301, 1304, 1305, 1308, 1312, 1313, 1316, 1317, 1320, 1321, 1324, 1325, 1328, 1332, 1336, 1340, 1344, 1348, 1352, 1354, 1356, 1358, 1360, 1362, 1364, 1366, 1368, 1372, 1375, 1376, 1380, 1384, 1388, 1392, 1396, 1400, 1404, 1408, 1412, 1415, 1416, 1420, 1424, 1428, 1432, 1435, 1436, 1440, 1444, 1448, 1452, 1456, 1459, 1460, 1464, 1468, 1472, 1476, 1480, 1484, 1488, 1492, 1496, 1499, 1500, 1504, 1508, 1512, 1516, 1519, 1520, 1524, 1528, 1532, 1536, 1540, 1543, 1544, 1548, 1552, 1556, 1558, 1560, 1562, 1564, 1566, 1568, 1570, 1572, 1576, 1579, 1580, 1584, 1588, 1592, 1596, 1600, 1604, 1608, 1612, 1616, 1620, 1624, 1628, 1632, 1636, 1640, 1644, 1648, 1652, 1656, 1660, 1664, 1668, 1672, 1676, 1680, 1684, 1688, 1692, 1694, 1696, 1698, 1700, 1702, 1704, 1708, 1712, 1716, 1720, 1724, 1728, 1732, 1736, 1740, 1744, 1748, 1752, 1756, 1760, 1764, 1768, 1772, 1776, 1780, 1784, 1788, 1792, 1796, 1800, 1804, 1808, 1810, 1812, 1814, 1816, 1818, 1820, 1824, 1828, 1832, 1836, 1840, 1844, 1848, 1852, 1856, 1860, 1864, 1868, 1872, 1876, 1880, 1884, 1888, 1892, 1896, 1900, 1904, 1908, 1912, 1916, 1920, 1924, 1928, 1932, 1936, 1938, 1940, 1942, 1944, 1948, 1952, 1956, 1960, 1964, 1968, 1972, 1976, 1980, 1984, 1988, 1992, 1996, 2000, 2004, 2008, 2012, 2016, 2020, 2024, 2028, 2032, 2036, 2040, 2044, 2048, 2050, 2052, 2054, 2056, 2060, 2064, 2068, 2072, 2076, 2080, 2084, 2088, 2092, 2096, 2100, 2104, 2108, 2112, 2116, 2120, 2124, 2128, 2132, 2136, 2140, 2144, 2148, 2150, 2152, 2154, 2156, 2158, 2160, 2162, 2164, 2166, 2168, 2172, 2176, 2180, 2184, 2188, 2192, 2196, 2200, 2204, 2208, 2212, 2216, 2220, 2224, 2228, 2232, 2236, 2240, 2244, 2248, 2252, 2256, 2260, 2262, 2264, 2266, 2268, 2272, 2276, 2280, 2284, 2288, 2292, 2296, 2300, 2304, 2308, 2312, 2316, 2320, 2324, 2328, 2332, 2336, 2340, 2344, 2348, 2352, 2370, 2374, 2465, 2469, 2473, 2477, 2478, 2481, 2482, 2485, 2489, 2493, 2497, 2501, 2505, 2509, 2513, 2517, 2521, 2525, 2529, 2533, 2537, 2541, 2545, 2549, 2553, 2557, 2561, 2565, 2569, 2573, 2577, 2581, 2598, 2602, 2706, 2710, 2779, 2803, 2814, 2818, 2823, 2843, 2863, 2883, 2898, 2910, 2914, 2938, 2942, 2956, 10000, 2965, 10001, 2969, 2978, 10002, 2987, 10003, 2996, 10004, 3005, 10005, 3012, 3016, 3021, 3026, 10006, 3035, 10007, 3044, 10008, 3053, 10009, 3061, 3066, 10010, 3069, 3079, 10011, 3082, 3083, 3096, 10012, 3099, 3106, 3110, 3111, 3114, 3115, 3116, 3118, 3119, 3122, 3123, 3126, 3127, 3130, 3131, 3134, 3135, 3138, 3142, 3146, 3150, 3154, 3158, 3160, 3162, 3166, 3170, 3174, 3178, 3182, 3186, 3190, 3194, 3198, 3202, 3206, 3210, 3214, 3218, 3222, 3226, 3230, 3234, 3247, 3251, 3255, 3280, 3293, 10013, 3296, 3306, 10014, 3309, 3317, 3323, 10015, 3326, 3336, 10016, 3339, 3349, 10017, 3352, 3361, 3366, 10018, 3369, 3379, 10019, 3382, 3391, 3396, 10020, 3405, 10021, 3409, 3416, 3421, 3426, 10022, 3435, 10023, 3444, 10024, 3453, 10025, 3461, 3466, 10026, 3475, 10027, 3484, 10028, 3493, 10029, 3501, 3506, 10030, 3513, 3517, 3523, 10031, 3527, 3536, 10032, 3545, 10033, 3554, 10034, 3563, 10035, 3571, 3576, 10036, 3585, 10037, 3594, 10038, 3603, 10039, 3610, 3614, 3618, 3624, 10040, 3633, 10041, 3637, 3646, 10042, 3655, 10043, 3664, 10044, 3673, 10045, 3681, 3686, 10046, 3695, 10047, 3704, 10048, 3711, 3712, 3715, 3719, 3723, 3733, 10049, 3741, 3746, 10050, 3755, 10051, 3759, 3768, 10052, 3777, 10053, 3786, 10054, 3795, 10055, 3804, 10056, 3811, 3812, 3815, 3819, 3823,...

Sequence for D=2

2, 20, 1, 6, 21, 9, 22, 10, 15, 23, 19, 222, 220, 25, 27, 29, 32, 35, 200, 40, 201, 45, 202, 47, 52, 203, 51, 59, 204, 64, 205, 69, 206, 74, 207, 79, 208, 84, 209, 89, 210, 94, 211, 99, 212, 101, 108, 213, 114, 214, 120, 215, 118, 127, 132, 216, 131, 141, 217, 147, 218, 152, 156, 219, 162, 221, 161, 163, 174, 223, 175, 182, 186, 224, 187, 195, 225, 232, 2222, 22222, 222222, 2222222, 2220, 196, 242, 198, 230, 236, 239, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 274, 275, 278, 281, 284, 287, 290, 293, 296, 299, 302, 307, 314, 2000, 321, 2001, 319, 329, 334, 2002, 337, 344, 2003, 351, 2004, 358, 2005, 365, 2006, 372, 2007, 371, 382, 2008, 381, 392, 2009, 391, 402, 2010, 401, 412, 2011, 411, 420, 423, 426, 429, 434, 2012, 437, 444, 2013, 451, 2014, 458, 2015, 465, 2016, 472, 2017, 471, 482, 2018, 481, 492, 2019, 491, 502, 2020, 501, 504, 515, 2021, 517, 523, 526, 529, 534, 2022, 536, 537, 547, 2023, 549, 557, 2024, 559, 567, 2025, 569, 577, 2026, 579, 587, 2027, 589, 597, 2028, 599, 607, 2029, 609, 617, 2030, 622, 623, 625, 628, 631, 639, 2031, 646, 2032, 649, 656, 2033, 662, 666, 2034, 672, 676, 2035, 682, 686, 2036, 692, 696, 2037, 702, 706, 2038, 712, 716, 2039, 721, 724, 727, 732, 2040, 731, 742, 2041, 741, 752, 2042, 751, 755, 765, 2043, 772, 2044, 771, 782, 2045, 781, 792, 2046, 791, 802, 2047, 801, 812, 2048, 811, 820, 823, 826, 829, 834, 2049, 841, 2050, 848, 2051, 855, 2052, 858, 865, 2053, 872, 2054, 871, 882, 2055, 881, 892, 2056, 891, 902, 2057, 901, 912, 2058, 911, 920, 923, 926, 929, 934, 2059, 941, 2060, 948, 2061, 955, 2062, 958, 965, 2063, 972, 2064, 971, 982, 2065, 981, 992, 2066, 991, 1002, ...

Sequence for D=3

2, 3, 30, 6, 31, 9, 32, 13, 33, 12, 14, 21, 34, 25, 35, 29, 333, 3333, 36, 38, 42, 300, 47, 301, 52, 302, 57, 303, 59, 63, 66, 304, 71, 305, 76, 306, 81, 307, 86, 308, 91, 309, 96, 310, 102, 311, 108, 312, 113, 117, 313, 119, 126, 314, 130, 133, 134, 136, 139, 142, 150, 315, 156, 316, 162, 317, 168, 318, 173, 177, 319, 183, 320, 182, 192, 321, 198, 322, 203, 207, 323, 209, 216, 324, 222, 325, 228, 326, 232, 235, 238, 243, 327, 242, 252, 328, 258, 329, 263, 267, 330, 268, 276, 331, 277, 285, 332, 286, 293, 297, 334, 33333, 333333, 3333333, 33333333, 333333333, 298, 338, 339, 341, 342, 344, 347, 350, 353, 356, 359, 361, 362, 365, 368, 371, 374, 377, 380, 383, 386, 389, 392, 394, 395, 398, 401, 404, 407, 410, 428, 3000, 433, 434, 436, 439, 442, 450, 3001, 457, 3002, 463, 467, 3003, 470, 477, 3004, 483, 487, 3005, 493, 497, 3006, 503, 507, 3007, 513, 517, 3008, 523, 527, 3009, 532, 535, 538, 543, 3010, 542, 553, 3011, 552, 563, 3012, 562, 573, 3013, 572, 576, 586, 3014, 593, 3015, 592, 603, 3016, 602, 613, 3017, 612, 623, 3018, 622, 631, 634, 637, 642, 3019, 649, 3020, 656, 3021, 663, 3022, 662, 673, 3023, 672, 676, 686, 3024, 693, 3025, 692, 703, 3026, 702, 713, 3027, 712, 723, 3028, 722, 731, 734, 737, 742, 3029, 749, 3030, 751, 759, 3031, 761, 769, 3032, 771, 779, 3033, 781, 782, 792, 3034, 794, 802, 3035, 804, 812, 3036, 814, 822, 3037, 824, 830, 833, 834, 836, 839, 842, 850, 3038, 852, 860, 3039, 862, 870, 3040, 877, 3041, 883, 887, 3042, 893, 897, 3043, 900, 907, 3044, 913, 917, 3045, 923, 927, 3046, 932, 935, 938, 943, 3047, 942, 953, 3048, 952, 963, 3049, 962, 973, 3050, 972, 983, 3051, 982, 993, 3052, 992, ...

Sequence for D=4

2, 4, 5, 44, 7, 40, 11, 41, 14, 17, 42, 21, 43, 24, 27, 45, 31, 46, 34, 37, 47, 54, 444, 4444, 48, 52, 404, 57, 400, 62, 401, 67, 402, 72, 403, 77, 405, 82, 406, 87, 407, 92, 408, 97, 409, 103, 410, 109, 411, 114, 118, 412, 124, 413, 123, 133, 414, 135, 140, 143, 146, 149, 154, 415, 153, 163, 416, 169, 417, 174, 178, 418, 184, 419, 183, 193, 420, 199, 421, 204, 208, 422, 214, 423, 213, 223, 424, 225, 232, 425, 238, 426, 242, 245, 248, 253, 427, 259, 428, 264, 268, 429, 274, 430, 273, 283, 431, 289, 432, 294, 298, 433, 304, 434, 303, 306, 316, 435, 322, 436, 328, 437, 334, 438, 333, 341, 344, 345, 347, 350, 358, 439, 364, 440, 363, 365, 376, 441, 377, 384, 388, 442, 389, 397, 443, 44444, 444444, 4444444, 44444444, 444444444, 4444444444, 398, 448, 449, 451, 452, 454, 457, 460, 462, 463, 466, 469, 472, 475, 478, 481, 484, 487, 490, 493, 495, 496, 499, 502, 505, 508, 511, 529, 4000, 536, 4001, 541, 544, 545, 547, 550, 558, 4002, 564, 568, 4003, 574, 578, 4004, 581, 588, 4005, 594, 598, 4006, 604, 608, 4007, 614, 618, 4008, 624, 628, 4009, 634, 638, 4010, 643, 646, 649, 654, 4011, 653, 664, 4012, 663, 674, 4013, 673, 684, 4014, 683, 687, 697, 4015, 704, 4016, 703, 714, 4017, 713, 724, 4018, 723, 734, 4019, 733, 742, 745, 748, 753, 4020, 760, 4021, 767, 4022, 774, 4023, 773, 784, 4024, 783, 787, 797, 4025, 804, 4026, 803, 814, 4027, 813, 824, 4028, 823, 834, 4029, 833, 842, 845, 848, 853, 4030, 860, 4031, 867, 4032, 874, 4033, 873, 884, 4034, 883, 887, 897, 4035, 904, 4036, 903, 914, 4037, 913, 924, 4038, 923, 934, 4039, 933, 942, 945, 948, 953, 4040, 955, 963, 4041, 965, 973, 4042, 975, 983, 4043, 985, 993, 4044, 995, ...

Sequence for D=5

2, 5, 4, 55, 7, 50, 11, 51, 15, 52, 14, 21, 53, 25, 54, 24, 31, 56, 35, 57, 34, 41, 58, 45, 59, 44, 65, 555, 5555, 550, 63, 505, 68, 500, 73, 501, 78, 502, 83, 503, 88, 504, 93, 506, 98, 507, 104, 508, 110, 509, 115, 119, 510, 125, 511, 124, 134, 512, 140, 513, 145, 149, 514, 153, 156, 159, 164, 515, 166, 173, 516, 179, 517, 185, 518, 184, 194, 519, 200, 520, 205, 209, 521, 215, 522, 214, 224, 523, 230, 524, 235, 239, 525, 241, 248, 526, 252, 255, 256, 258, 261, 269, 527, 275, 528, 274, 284, 529, 290, 530, 295, 299, 531, 305, 532, 304, 314, 533, 320, 534, 325, 329, 535, 331, 338, 536, 344, 537, 350, 538, 348, 357, 362, 539, 368, 540, 374, 541, 380, 542, 385, 389, 543, 395, 544, 394, 404, 545, 406, 413, 546, 419, 547, 425, 548, 424, 434, 549, 440, 551, 441, 449, 552, 450, 453, 456, 459, 462, 470, 553, 471, 479, 554, 480, 488, 556, 489, 497, 557, 55555, 555555, 5555555, 55555555, 555555555, 5555555555, 55555555555, 559, 562, 565, 564, 498, 571, 574, 577, 580, 583, 586, 589, 592, 595, 597, 598, 601, 610, 5000, 617, 5001, 624, 5002, 631, 5003, 638, 5004, 645, 5005, 644, 648, 656, 659, 664, 5006, 671, 5007, 678, 5008, 685, 5009, 684, 695, 5010, 694, 705, 5011, 704, 715, 5012, 714, 725, 5013, 724, 735, 5014, 734, 745, 5015, 744, 748, 756, 759, 764, 5016, 771, 5017, 778, 5018, 785, 5019, 784, 795, 5020, 794, 805, 5021, 804, 815, 5022, 814, 825, 5023, 824, 835, 5024, 834, 845, 5025, 844, 848, 856, 859, 864, 5026, 871, 5027, 878, 5028, 885, 5029, 884, 895, 5030, 894, 905, 5031, 904, 915, 5032, 914, 925, 5033, 924, 935, 5034, 934, 945, 5035, 944, 948, 956, 959, 964, 5036, 971, 5037, 978, 5038, 985, 5039, 984, 995, 5040, 994, ...

Sequence for D=6

2, 6, 4, 60, 61, 9, 62, 13, 63, 16, 19, 64, 23, 65, 26, 29, 66, 30, 35, 67, 39, 68, 43, 69, 46, 49, 600, 54, 601, 59, 666, 6666, 66660, 73, 602, 78, 603, 83, 604, 88, 605, 93, 606, 95, 101, 607, 106, 110, 608, 116, 609, 115, 125, 610, 131, 611, 136, 140, 612, 146, 613, 145, 155, 614, 161, 615, 159, 168, 173, 616, 175, 182, 617, 188, 618, 194, 619, 200, 620, 206, 621, 205, 215, 622, 221, 623, 226, 230, 624, 236, 625, 235, 245, 626, 247, 254, 627, 260, 628, 258, 267, 272, 629, 278, 630, 284, 631, 290, 632, 296, 633, 295, 305, 634, 311, 635, 316, 320, 636, 322, 329, 637, 335, 638, 341, 639, 346, 350, 640, 356, 641, 355, 363, 366, 367, 369, 372, 380, 642, 386, 643, 385, 395, 644, 401, 645, 406, 410, 646, 412, 419, 647, 425, 648, 431, 649, 436, 440, 650, 446, 651, 445, 455, 652, 461, 653, 459, 468, 473, 654, 479, 655, 485, 656, 487, 494, 657, 500, 658, 506, 659, 505, 515, 660, 516, 520, 526, 530, 661, 531, 539, 662, 540, 548, 663, 549, 556, 560, 664, 558, 561, 567, 570, 578, 665, 579, 586, 590, 667, 591, 599, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 668, 670, 673, 676, 678, 679, 682, 685, 688, 691, 694, 697, 700, 709, 6000, 716, 6001, 715, 726, 6002, 725, 736, 6003, 735, 746, 6004, 745, 756, 6005, 755, 764, 767, 772, 6006, 775, 782, 6007, 789, 6008, 796, 6009, 795, 806, 6010, 805, 816, 6011, 815, 826, 6012, 825, 836, 6013, 835, 846, 6014, 845, 856, 6015, 855, 864, 867, 872, 6016, 875, 882, 6017, 889, 6018, 896, 6019, 895, 906, 6020, 905, 916, 6021, 915, 926, 6022, 925, 936, 6023, 935, 946, 6024, 945, 956, 6025, 955, 964, 967, 972, 6026, 975, 982, 6027, 989, 6028, 996, 6029, ...

Sequence for D=7

2, 7, 4, 70, 8, 77, 11, 71, 15, 72, 19, 73, 23, 74, 27, 75, 26, 33, 76, 37, 78, 36, 43, 79, 47, 700, 46, 54, 701, 59, 702, 64, 703, 69, 777, 7777, 77770, 83, 704, 87, 90, 705, 95, 706, 101, 707, 103, 110, 708, 116, 709, 122, 710, 127, 131, 711, 137, 712, 136, 146, 713, 152, 714, 157, 161, 715, 167, 716, 166, 174, 177, 178, 180, 187, 191, 717, 193, 200, 718, 206, 719, 212, 720, 217, 221, 721, 227, 722, 226, 236, 723, 242, 724, 247, 251, 725, 257, 726, 256, 266, 727, 268, 273, 276, 279, 284, 728, 290, 729, 296, 730, 302, 731, 307, 311, 732, 317, 733, 316, 326, 734, 332, 735, 337, 341, 736, 347, 737, 346, 349, 359, 738, 365, 739, 371, 740, 369, 378, 383, 741, 389, 742, 395, 743, 401, 744, 407, 745, 406, 416, 746, 422, 747, 424, 431, 748, 437, 749, 436, 446, 750, 452, 751, 457, 461, 752, 467, 753, 466, 474, 477, 478, 480, 487, 491, 754, 497, 755, 496, 506, 756, 512, 757, 514, 521, 758, 527, 759, 526, 536, 760, 542, 761, 547, 551, 762, 557, 763, 556, 566, 764, 570, 573, 576, 579, 584, 765, 590, 766, 596, 767, 598, 605, 768, 611, 769, 617, 770, 616, 618, 629, 771, 630, 637, 641, 772, 642, 650, 773, 651, 659, 774, 660, 667, 671, 775, 669, 672, 678, 681, 689, 776, 690, 697, 787, 77777, 777777, 7777777, 77777777, 777777777, 7777777777, 77777777777, 777777777777, 7777777770, 698, 782, 797, 785, 788, 791, 794, 803, 7000, 810, 7001, 817, 7002, 816, 827, 7003, 826, 837, 7004, 836, 847, 7005, 846, 857, 7006, 856, 867, 7007, 866, 870, 875, 878, 881, 889, 7008, 896, 7009, 903, 7010, 910, 7011, 917, 7012, 916, 927, 7013, 926, 937, 7014, 936, 947, 7015, 946, 957, 7016, 956, 967, 7017, 966, 970, 975, 978, 981, 989, 7018, 996, 7019, ...

Sequence for D=8

2, 8, 4, 80, 7, 88, 11, 81, 15, 82, 18, 21, 83, 25, 84, 28, 31, 85, 35, 86, 38, 41, 87, 45, 89, 48, 51, 800, 56, 801, 61, 802, 66, 803, 71, 804, 76, 805, 98, 888, 8888, 880, 93, 806, 99, 881, 104, 807, 110, 808, 112, 118, 122, 809, 128, 810, 127, 137, 811, 143, 812, 148, 152, 813, 158, 814, 157, 167, 815, 173, 816, 178, 180, 183, 186, 189, 194, 817, 200, 818, 202, 208, 212, 819, 218, 820, 217, 227, 821, 233, 822, 238, 242, 823, 248, 824, 247, 257, 825, 263, 826, 268, 272, 827, 278, 828, 277, 280, 285, 288, 291, 292, 302, 829, 308, 830, 307, 317, 831, 323, 832, 328, 332, 833, 338, 834, 337, 347, 835, 353, 836, 358, 362, 837, 368, 838, 367, 370, 380, 839, 378, 385, 387, 390, 398, 840, 397, 407, 841, 413, 842, 418, 422, 843, 428, 844, 427, 437, 845, 443, 846, 448, 452, 847, 458, 848, 457, 460, 470, 849, 476, 850, 480, 483, 486, 489, 494, 851, 500, 852, 506, 853, 512, 854, 518, 855, 517, 527, 856, 533, 857, 538, 542, 858, 544, 551, 859, 557, 860, 563, 861, 568, 572, 862, 578, 863, 577, 585, 588, 589, 591, 598, 602, 864, 608, 865, 607, 617, 866, 623, 867, 628, 632, 868, 634, 641, 869, 647, 870, 653, 871, 658, 662, 872, 668, 873, 667, 677, 874, 681, 684, 687, 692, 875, 698, 876, 697, 707, 877, 713, 878, 715, 722, 879, 728, 882, 727, 729, 740, 883, 741, 748, 752, 884, 753, 761, 885, 762, 770, 886, 771, 778, 780, 783, 786, 789, 794, 887, 795, 88888, 888888, 8888888, 88888888, 888888888, 8888888888, 88888888888, 888888888888, 8888888888888, 888888880, 890, 893, 896, 899, 905, 8000, 912, 8001, 918, 922, 8002, 928, 932, 8003, 938, 942, 8004, 948, 952, 8005, 958, 962, 8006, 968, 972, 8007, 978, 980, 983, 986, 989, 994, 8008, 997, ...

Sequence for D=9

2, 9, 4, 90, 7, 91, 92, 13, 93, 17, 94, 21, 95, 25, 96, 29, 97, 28, 35, 98, 39, 99, 38, 40, 47, 900, 52, 901, 57, 902, 62, 903, 67, 904, 72, 905, 77, 906, 82, 907, 87, 908, 999, 9999, 9990, 104, 909, 106, 113, 910, 119, 911, 118, 128, 912, 134, 913, 139, 143, 914, 149, 915, 148, 158, 916, 164, 917, 169, 173, 918, 179, 919, 178, 181, 191, 920, 189, 196, 198, 201, 209, 921, 208, 218, 922, 224, 923, 229, 233, 924, 239, 925, 238, 248, 926, 254, 927, 259, 263, 928, 269, 929, 268, 271, 281, 930, 287, 931, 291, 294, 297, 302, 932, 308, 933, 314, 934, 319, 323, 935, 329, 936, 328, 338, 937, 344, 938, 349, 353, 939, 355, 362, 940, 368, 941, 374, 942, 379, 383, 943, 389, 944, 388, 396, 399, 400, 407, 945, 413, 946, 419, 947, 418, 428, 948, 434, 949, 436, 443, 950, 449, 951, 448, 458, 952, 464, 953, 469, 473, 954, 479, 955, 478, 488, 956, 492, 495, 498, 503, 957, 509, 958, 508, 518, 959, 520, 527, 960, 533, 961, 539, 962, 538, 548, 963, 554, 964, 559, 563, 965, 569, 966, 568, 578, 967, 584, 968, 589, 591, 594, 597, 602, 969, 604, 611, 970, 617, 971, 623, 972, 629, 973, 628, 638, 974, 644, 975, 649, 653, 976, 659, 977, 658, 668, 978, 674, 979, 676, 683, 980, 689, 981, 688, 696, 699, 700, 707, 982, 713, 983, 719, 984, 718, 728, 985, 734, 986, 739, 743, 987, 749, 988, 748, 758, 989, 760, 767, 990, 768, 776, 991, 777, 785, 992, 786, 792, 795, 798, 803, 993, 804, 812, 994, 813, 821, 995, 822, 829, 833, 996, 834, 842, 997, 843, 851, 998, 852, 859, 863, 9000, 869, 873, 9001, 879, 883, 9002, 889, 891, 894, 897, 1999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 9900, ...

Sequence for D=0 (999 terms kindly computed by Mehdi Tibouchi, on december 3rd, 2009, showing 915 digits “0”):

3, 10, 6, 20, 9, 100, 14, 30, 18, 40, 50, 24, 60, 28, 70, 80, 34, 90, 38, 1000, 39, 46, 101, 110, 54, 102, 59, 200, 64, 103, 69, 300, 74, 104, 79, 400, 84, 105, 89, 500, 94, 106, 99, 100000000, 1010, 108, 112, 121, 201, 127, 202, 133, 203, 139, 204, 145, 205, 151, 206, 157, 207, 163, 208, 169, 209, 175, 301, 181, 302, 187, 303, 193, 304, 199, 100000000000, 214, 305, 220, 306, 218, 229, 307, 235, 308, 241, 309, 247, 401, 253, 402, 259, 403, 265, 404, 271, 405, 277, 406, 283, 407, 289, 408, 295, 409, 1000000000000, 298, 299, 320, 501, 318, 329, 502, 335, 503, 341, 504, 347, 505, 353, 506, 359, 507, 365, 508, 371, 509, 377, 600, 378, 386, 601, 390, 395, 602, 1010000000000, 398, 417, 603, 423, 604, 429, 605, 435, 606, 441, 607, 447, 608, 453, 609, 459, 700, 460, 463, 471, 701, 477, 702, 483, 703, 489, 704, 495, 705, 1020000000000, 498, 517, 706, 523, 707, 529, 708, 535, 709, 541, 800, 542, 550, 801, 548, 559, 802, 565, 803, 571, 804, 577, 805, 583, 806, 589, 807, 595, 808, 1030000000000, 598, 617, 809, 623, 900, 624, 630, 635, 901, 641, 902, 647, 903, 653, 904, 659, 905, 665, 906, 671, 907, 677, 908, 683, 909, 689, 1001, 690, 694, 110000000000, 714, 1002, 715, 724, 1003, 725, 734, 1004, 735, 744, 1005, 745, 754, 1006, 755, 764, 1007, 765, 774, 1008, 775, 784, 1009, 785, 794, 1011, 1040000000000, 798, 817, 1012, 824, 1013, 831, 1014, 838, 1015, 845, 1016, 850, 855, 1017, 860, 865, 1018, 870, 875, 1019, 880, 885, 1020, 887, 895, 1021, 120000000000, 914, 1022, 921, 1023, 928, 1024, 935, 1025, 940, 945, 1026, 950, 955, 1027, 960, 965, 1028, 970, 975, 1029, 980, 985, 1030, 987, 995, 1031, 1000000000000000000000000000000000, 999, 1036, 1040, 1042, 1044, 1048, 1052, 1056, 1060, 1064, 1066, 1068, 1072, 1076, 1080, 1084, 1088, 1090, 1092, 1096, 1100, 1102, 1104, 1108, 1113, 1114, 1117, 1121, 1125, 1150, 1156, 2000, 1157, 1158, 1170, 1176, 2001, 1177, 1188, 2002, 1189, 1200, 2003, 1197, 1198, 1201, 1213, 1224, 2004, 1225, 1236, 2005, 1237, 1248, 2006, 1249, 1260, 2007, 1258, 1261, 1276, 2008, 1277, 1288, 2009, 1289, 1300, 2010, 1297, 1298, 1302, 1313, 1324, 2011, 1330, 1336, 2012, 1344, 2013, 1350, 1356, 2014, 1364, 2015, 1370, 1376, 2016, 1384, 2017, 1390, 1396, 2018, 1401, 1405, 1409, 1416, 2019, 1424, 2020, 1426, 1436, 2021, 1444, 2022, 1450, 1456, 2023, 1464, 2024, 1470, 1476, 2025, 1484, 2026, 1490, 1496, 2027, 1501, 1505, 1509, 1516, 2028, 1524, 2029, 1530, 1536, 2030, 1538, 1548, 2031, 1556, 2032, 1564, 2033, 1570, 1576, 2034, 1584, 2035, 1590, 1596, 2036, 1601, 1605, 1609, 1616, 2037, 1624, 2038, 1630, 1636, 2039, 1644, 2040, 1646, 1656, 2041, 1664, 2042, 1670, 1676, 2043, 1684, 2044, 1690, 1696, 2045, 1701, 1705, 1709, 1716, 2046, 1724, 2047, 1730, 1736, 2048, 1744, 2049, 1750, 1756, 2050, 1758, 1768, 2051, 1776, 2052, 1784, 2053, 1790, 1796, 2054, 1801, 1805, 1809, 1816, 2055, 1824, 2056, 1830, 1836, 2057, 1844, 2058, 1850, 1856, 2059, 1864, 2060, 1866, 1876, 2061, 1884, 2062, 1890, 1896, 2063, 1901, 1905, 1909, 1916, 2064, 1924, 2065, 1930, 1936, 2066, 1944, 2067, 1950, 1956, 2068, 1964, 2069, 1970, 1976, 2070, 1978, 1988, 2071, 1996, 2072, 10000000000000000000000000000000000000000000000000000000000000000000000000, 2074, 2078, 2082, 2086, 2090, 2092, 2094, 2098, 2102, 2107, 2111, 2120, 2126, 3000, 2127, 2128, 2140, 2146, 3001, 2147, 2158, 3002, 2159, 2170, 3003, 2168, 2171, 2186, 3004, 2187, 2198, 3005, 2199, 2207, 2214, 3006, 2215, 2226, 3007, 2227, 2238, 3008, 2239, 2250, 3009, 2248, 2251, 2266, 3010, 2268, 2278, 3011, 2286, 3012, 2294, 3013, 2300, 2299, 2307, 2314, 3014, 2320, 2326, 3015, 2334, 3016, 2340, 2346, 3017, 2354, 3018, 2360, 2366, 3019, 2374, 3020, 2376, 2386, 3021, 2394, 3022, 2400, 2399, 2407, 2414, 3023, 2420, 2426, 3024, 2434, 3025, 2440, 2446, 3026, 2454, 3027, 2460, 2466, 3028, 2474, 3029, 2480, 2486, 3030, 2488, 2498, 3031, 2503, 2507, 2514, 3032, 2520, 2526, 3033, 2534, 3034, 2540, 2546, 3035, 2554, 3036, 2560, 2566, 3037, 2574, 3038, 2580, 2586, 3039, 2594, 3040, 2596, 2603, 2607, 2614, 3041, 2620, 2626, 3042, 2634, 3043, 2640, 2646, 3044, 2654, 3045, 2660, 2666, 3046, 2674, 3047, 2680, 2686, 3048, 2694, 3049, 2700, 2699, 2707, 2714, 3050, 2716, 2726, 3051, 2734, 3052, 2740, 2746, 3053, 2754, 3054, 2760, 2766, 3055, 2774, 3056, 2780, 2786, 3057, 2794, 3058, 2800, 2799, 2807, 2814, 3059, 2820, 2826, 3060, 2828, 2838, 3061, 2846, 3062, 2854, 3063, 2860, 2866, 3064, 2874, 3065, 2880, 2886, 3066, 2894, 3067, 2900, 2899, 2907, 2914, 3068, 2920, 2926, 3069, 2934, 3070, 2936, 2946, 3071, 2954, 3072, 2960, 2966, 3073, 2974, 3074, 2980, 2986, 3075, 2994, 3076, 10000000000000000000000000000000000000000000000000000000000000000000000000000000, 2998, 2999, 3086, 3090, 3092, 3094, 3098, 3102, 3107, 3111, 3120, 3126, 4000, 3127, 3128, 3140, 3146, 4001, 3147, 3158, 4002, 3159, 3170, 4003, 3168, 3171, 3186, 4004, 3187, 3198, 4005, 3199, 3207, 3214, 4006, 3215, 3226, 4007, 3227, 3238, 4008, 3239, 3250, 4009, 3248, 3251, 3266, 4010, 3268, 3278, 4011, 3286, 4012, 3294, 4013, 3300, 3299, 3307, 3314, 4014, 3320, 3326, 4015, 3334, 4016, 3340, 3346, 4017, 3354, 4018, 3360, 3366, 4019, 3374, 4020, 3376, 3386, 4021, 3394, 4022, 3400, 3399, 3407, 3414, 4023, 3420, 3426, 4024, 3434, 4025, 3440, 3446, 4026, 3454, 4027, 3460, 3466, 4028, 3474, 4029, 3480, 3486, 4030, 3488, 3498, 4031, 3503, 3507, 3514, 4032, 3520, 3526, 4033, 3534, 4034, 3540, 3546, 4035, 3554, 4036, 3560, 3566, 4037, 3574, 4038, 3580, 3586, 4039, 3594, 4040, 3596, 3603, 3607, 3614, 4041, 3620, 3626, 4042, 3634, 4043, 3640, 3646, 4044, 3654, 4045, 3660, 3666, 4046, 3674, 4047, 3680, 3686, 4048, 3694, 4049, 3700, 3699, 3707, 3714, 4050, 3716, 3726, 4051, 3734, 4052, 3740, 3746, 4053, 3754, 4054, 3760, 3766, 4055, 3774, 4056, 3780, 3786, 4057, 3794, 4058, 3800, 3799, 3807, 3814, 4059, 3820, 3826, 4060, 3828, 3838, 4061, 3846, 4062, 3854, 4063, 3860, 3866, 4064, 3874, 4065, 3880, 3886, 4066, 3894, 4067, 3900, 3899, 3907, 3914, 4068, 3920, 3926, 4069, 3934, 4070, 3936, 3946, 4071,...

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Merci beaucoup, “zwim” et Mehdi ! Ça a l’air correct de bout en bout ; je vais proposer ces suites à l’OEIS début décembre avec votre nom comme co-auteur !

Et merci aussi à Jean-Paul Davalan qui a écrit deux scripts permettant de calculer ces suites :

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