Une merveille avec a(2n)

 

Une merveille avec a(2n)

Le tableau ci-dessous reprend pour chaque n la somme des chiffres impairs utilisés jusque-là dans S (Σ imp.) et la somme des chiffres pairs (Σ pairs).

S = 1, 2, 3, 6, 4, 11, 5, 28, 7, 68, 8, 13, 9, 288, 10, 14, 12, 36, 15, 48, 16, 23, 17, 88, 19, …

On voit que pour a(2n) la somme des chiffres pairs utilisés par S jusque-là vaut le double de la somme des chiffres impairs.

Exemple :

pour n = 1 nous avons a(2n) = a(2) = 2, la Σ pairs qui vaut 2 et la Σ imp. qui vaut 1 (la moitié) ;

pour n = 2 nous avons a(2n) = a(4) = 6, la Σ pairs qui vaut 8 et la Σ imp. qui vaut 4 (la moitié) ;

pour n = 3 nous avons a(2n) = a(6) = 11, la Σ pairs qui vaut 12 et la Σ imp. qui vaut 6 (la moitié) ;

pour n = 4 nous avons a(2n) = a(8) = 28, la Σ pairs qui vaut 22 et la Σ imp. qui vaut 11 (la moitié) ; etc.

S est un cauchemar à calculer à la main ! Nous ne garantissons pas son exactitude...

Il faut, avant de proposer le terme suivant, tester tous les plus petits que lui qui ne sont pas encore dans S. Ainsi sera-t-on sûr que S est bien la lexico-première suite obéissant à cette définition :

« La somme des chiffres pairs qui figurent dans la suite jusque et y compris a(2n) vaut le double de la somme des chiffres impairs utilisés jusque-là. »

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Mise à jour du 9 février 2022 à midi

Carole D. fut ultraluminique et corrigea illico ma suite (mon 18e terme, ci-dessus, était faux) : 

S = 1, 2, 3, 6, 4, 11, 5, 28, 7, 68, 8, 13, 9, 288, 10, 14, 12, 21, 15, 48, 16, 23, 17, 88, 18, 30, 19, 488, 20, 34, 22, 32, 24, 45, 25, 26, 27, 66, 29, 268, 31, 44, 33, 84, 35, 286, 36, 63, 37, 668, 38, 43, 39, 888, 40, 56, 41, 58, 42, 54, 46, 47, 49, 86, 50, 64, 51, 168, 52, 62, 53, 448, 55, 686, 57, 2688, 59, 4888, 60, 78, 61, 65, 67, 80, 69, 186, 70, 188, 71, 466, 72, 228, 73, 848, 74, 82, 75, 2868, 76, 128, 77, 6688, 79, 8888, 81, 87, 83, 85, 89, 148, 90, 468, 91, 866, 92, 484, 93, 2886, 94, 248, 95, 6868, 96, 246, 97, 26888, 98, 166, 99, 48888, 100, 104, 101, 106, 102, 114, 103, 146, 105, 264, 107, 628, 108, 111, 109, 884, 110, 118, 112, 116, 113, 184, 115, 266, 117, 486, 119, 688, 120, 138, 121, 122, 123, 126, 124, 134, 125, 208, 127, 284, 129, 648, 130, 164, 131, 226, 132, 144, 133, 428, 135, 666, 136, 140, 137, 868, 139, 2888, 141, 183, 142, 143, 145, 182, 147, 282, 149, 646, 150, 408, ...

Les sommes paires [0, 2, 2, 8, 12, 12, 12, 22, 22, 36, 44, 44, 44, 62, 62, 66, 68, 70, 70, 82, 88, 90, 90, 106, 114, 114, 114, 134, 136, 140, 144, 146, 152, 156, 158, 166, 168, 180, 182, 198, 198, 206, 206, 218, 218, 234, 240, 246, 246, 266, 274, 278, 278, 302, 306, 312, 316, 324, 330, 334, 344, 348, 352, 366, 366, 376, 376, 390, 392, 400, 400, 416, 416, 436, 436, 460, 460, 488, 494, 502, 508, 514, 520, 528, 534, 548, 548, 564, 564, 580, 582, 594, 594, 614, 618, 628, 628, 652, 658, 668, 668, 696, 696, 728, 736, 744, 752, 760, 768, 780, 780, 798, 798, 818, 820, 836, 836, 860, 864, 878, 878, 906, 912, 924, 924, 956, 964, 976, 976, 1012, 1012, 1016, 1016, 1022, 1024, 1028, 1028, 1038, 1038, 1050, 1050, 1066, 1074, 1074, 1074, 1094, 1094, 1102, 1104, 1110, 1110, 1122, 1122, 1136, 1136, 1154, 1154, 1176, 1178, 1186, 1188, 1192, 1194, 1202, 1208, 1212, 1214, 1224, 1226, 1240, 1242, 1260, 1260, 1270, 1270, 1280, 1282, 1290, 1290, 1304, 1304, 1322, 1328, 1332, 1332, 1354, 1354, 1380, 1384, 1392, 1398, 1402, 1406, 1416, 1420, 1432, 1436, 1452, 1452, 1464, ...] 

Les sommes impaires [1, 1, 4, 4, 4, 6, 11, 11, 18, 18, 18, 22, 31, 31, 32, 33, 34, 35, 41, 41, 42, 45, 53, 53, 54, 57, 67, 67, 67, 70, 70, 73, 73, 78, 83, 83, 90, 90, 99, 99, 103, 103, 109, 109, 117, 117, 120, 123, 133, 133, 136, 139, 151, 151, 151, 156, 157, 162, 162, 167, 167, 174, 183, 183, 188, 188, 194, 195, 200, 200, 208, 208, 218, 218, 230, 230, 244, 244, 244, 251, 252, 257, 264, 264, 273, 274, 281, 282, 290, 290, 297, 297, 307, 307, 314, 314, 326, 326, 333, 334, 348, 348, 364, 364, 365, 372, 375, 380, 389, 390, 399, 399, 409, 409, 418, 418, 430, 430, 439, 439, 453, 453, 462, 462, 478, 478, 487, 488, 506, 506, 507, 508, 510, 511, 512, 514, 518, 519, 525, 525, 533, 533, 534, 537, 547, 547, 549, 551, 553, 555, 560, 561, 568, 568, 577, 577, 588, 588, 589, 593, 595, 596, 600, 601, 602, 606, 612, 612, 620, 620, 630, 630, 634, 635, 640, 640, 644, 645, 652, 652, 661, 661, 665, 666, 677, 677, 690, 690, 692, 696, 697, 701, 707, 708, 716, 716, 726, 726, 732, 732, ...]

Je soumets S à l’OEIS dans quelques laps – merci Carole !

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Et voici la même idée « inversée » (re-merci Carole !) où c’est la somme cumulée des chiffres impairs qui vaut le double de la somme cumulée des chiffres pairs (suite proposée aussi à l’OEIS – bientôt ici) :

0, 123, 1, 23, 2, 13, 3, 12, 4, 17, 5, 34, 6, 39, 7, 14, 8, 79, 9, 36, 10, 32, 11, 112, 15, 114, 16, 119, 18, 159, 19, 116, 20, 31, 21, 27, 22, 35, 24, 57, 25, 47, 26, 97, 28, 1199, 29, 43, 30, 45, 33, 141, 37, 158, 38, 139, 40, 53, 41, 70, 42, 75, 44, 709, 46, 1379, 48, 1599, 49, 74, 50, 67, 51, 156, 52, 203, 54, 72, 55, 161, 56, 115, 58, 137, 59, 118, 60, 93, 61, 155, 62, 790, 63, 90, 64, 1397, 65, 133, 66, 1779, 68, 1999, 69, 111, 71, 136, 73, 185, 76, 92, 77, 181, 78, 117, 80, 907, 81, 177, 82, 1559, 83, 157, 84, 1797, 85, 173, 86, 3799, 87, 135, 88, 5999, 89, 151, 91, 338, 94, 227, 95, 811, 96, 207, 98, 313, 99, 1128, 100, 230, 101, 121, 102, 270, 103, 134, 104, 331, 105, 165, 106, 191, 107, 163, 108, 195, 109, 383, 110, 145, 113, 223, 120, 300, 122, 511, 124, 317, 125, 154, 126, 339, 127, 143, 128, 199, 129, 189, 130, 167, 131, 232, 132, 147, 138, 279, 140, 700, 142, 335, 144, 357, 146, 379, 148, 599, 149, 169, 150, 198, 152, 196, 153, 234, 160, 353, 162, 375, 164, 397, 166, 779, 168, 999, 170, 178, 171, 243, 172, 176, 174, 213, 175, 236, 179, 238, 180, 393, 182, 559, 183, 297, 184, 797, 186, 9099, 187, 239, 188, 13999, 190, 518, 192, 336, 193, 263, 194, 211, 197, 283, 200, 217, 201, 702, 202, 257, 204, 309, 205, 272, 206, 970, 208, 1577, 209, 247, 210, 720, 212, 1015, 214, 371, 215, 334, 216, 519, 218, 577, 219, 363, 220, 275, 221, 1033, 222, 390, 224, 1159, 225, 1011, 226, 1595, 228, 1959, 229, 302, 231, 312, 233, 343, 235, 314, 237, 378, 240, 507, 241, 515, 242, 1177, 244, 1739, 245, 1051, 246, 1977, 248, 3979, 249, 1101, 250, 320, 251, 367, 252, 1110, 253, 341, 254, 1105, 255, 387, 256, 533, 258, 537, 259, 581, 260, 1195, 261, 555, 262, 1757, 264, 1995, 265, 551, 266, 3997, 267, 315, 268, 7799, 269, 1129, 271, 356, 273, 411, 274, 290, 276, 333, 277, 611, 278, 319, 280, 1775, 281, 595, 282, 3399, 284, 5599, 285, 573, 286, 7979, 287, 337, 288, 9999, 289, 713, 291, 516, 292, 407, 293, 316, 294, 1125, 295, 815, 296, 1150, 298, 731, 299, 1018, 301, 365, 303, 558, 304, 311, 305, 358, 306, 351, 307, 833, 308, 355, 310, 389, 318, 570, 321, 345, 322, 500, 323, 376, 324, 513, 325, 398, 326, 373, 327, 561, 328, 359, 329, 361, 330, 585, 332, 415, 340, 902, 342, 531, 344, 391, 346, 377, 347, 433, 348, 399, 349, 413, 350, 385, 352, 431, 354, 369, 360, 711, 362, 517, 364, 395, 366, 579, 368, 799, 370, 851, 372, 615, 374, 451, 380, 535, 381, 705, 382, 539, 384, 597, 386, 979, 388, 11999, 392, 400, 394, 536, 396, 417, 401, 1192, 402, 729, 403, 920, 404, 1339, 405, 1149, 406, 1793, 408, 3579, 409, 470, 410, 1219, 412, 911, 414, 591, 416, 739, 418, 959, 419, 514, 420, 750, 421, 1019, 422, 1357, 423, 900, 424, 1919, 425, 1237, 426, 3597, 427, 1013, 428, 5779, 429, 1152, 430, 1031, 432, 1017, 434, 553, 435, 453, 436, 557, 437, 541, 438, 759, 439, 563, 440, 1375, 441, 717, 442, 1937, 443, 571, 444, 3759, 445, 1037, 446, 5797, 447, 1035, 448, 7997, 449, 1255, 450, 1194, 452, 1273, 454, 1055, 455, 556, 456, 735, 457, 578, 458, 757, 459, 633, 460, 1955, 461, 775, 462, 3777, 463, 575, 464, 5959, 465, 753, 466, 9599, 467, 715, 468, 90999, 469, 1073, 471, 534, 472, 1103, 473, 565, 474, 1053, 475, 587, 476, 733, 477, 651, 478, 593, 479, 538, 480, 3795, 481, 977, 482, 5977, 483, 777, 484, 9779, 485, 793, 486, 99099, 487, 719, 488, 139999, 489, 771, 490, 704, 491, 589, 492, 1215, 493, 617, 494, 1291, 495, 738, 496, 1091, 497, 583, 498, 915, 499, 1016, 501, 783, 502, 722, 503, 613, 504, 1233, 505, 1058, 506, 1303, 508, 1109, 509, 1081, 510, 819, 512, 598, 520, 740, 521, 619, 522, 1251, 523, 635, 524, 1327, 525, 837, 526, 1190, 527, 631, 528, 933, 529, 1061, 530, 718, 532, 653, 540, 1323, 542, 1330, 543, 567, 544, 1259, 545, 637, 546, 951, 547, 671, 548, 919, 549, 855, 550, 1085, 552, 873, 554, 655, 560, 1372, 562, 1277, 564, 1059, 566, 937, 568, 995, 569, 673, 572, 781, 574, 716, 576, 639, 580, 1295, 582, 1077, 584, 955, 586, 2999, 588, 9909, 590, 1108, 592, 1106, 594, 891, 596, 691, 600, 792, 601, 1307, 602, 1393, 603, 1071, 604, 1973, 605, 1459, 606, 3939, 607, 1127, 608, 5995, 609, 1332, 610, 1370, 612, 1095, 614, 973, 616, 5099, 618, 9990, 620, 1519, 621, 1299, 622, 1991, 623, 751, 624, 3957, 625, 1499, 626, 7399, 627, 1107, 628, 9797, 629, 1477, 630, 1170, 632, 913, 634, 737, 636, 795, 638, 997, 640, 3179, 641, 991, 642, 3975, 643, 755, 644, 7579, 645, 1509, 646, 9959, 647, 931, 648, 99909, 649, 1505, 650, 1495, 652, 1529, 654, 1590, 656, 1099, 657, 675, 658, 5909, 659, 736, 660, 3993, 661, 5990, 662, 7597, 663, 939, 664, 9977, 665, 1909, 666, 99990, 667, 773, 668, 157999, 669, 1707, 670, 1130, 672, 1239, 674, 1039, 676, 791, 677, 763, 678, 957, 679, 758, 680, 7759, 681, 11799, 682, 9995, 683, 7099, 684, 117999, 685, 7079, 686, 159799, 687, 975, 688, 179999, 689, 1990, 690, 1347, 692, 1501, 693, 714, 694, 1550, 695, 778, 696, 1770, 697, 761, 698, 2599, 699, 918, 701, 817, 703, 1124, 706, 1172, 707, 1126, 708, 1257, 710, 835, 712, 785, 721, 839, 723, 981, 724, 1217, 725, 853, 726, 1275, 727, 1142, 728, 1057, 730, 1160, 732, 1014, 734, 787, 741, 756, 742, 1235, 743, 859, 744, 1293, 745, 857, 746, 1075, 747, 1041, 748, 917, 749, 871, 752, 1036, 754, 875, 760, 1253, 762, 1305, 764, 1093, 765, 789, 766, 935, 767, 877, 768, 993, 769, 893, 770, 1144, 772, 1214, 774, 1056, 776, 895, 780, 1329, 782, 1279, 784, 953, 786, 2799, 788, 7909, 794, 1063, 796, 938, 798, 879, 800, 1537, 801, 1905, 802, 3197, 803, 1297, 804, 5199, 805, 1592, 806, 7777, 807, 1350, 808, 29999, 809, 1510, 810, 1929, 812, 2779, 813, 903, 814, 7097, 816, 11979, 818, 15799, 820, 3359, 821, 2797, 822, 5379, 823, 971, 824, 7795, 825, 1950, 826, 50999, 827, 1309, 828, 119799, 829, 1703, 830, 1390, 831, 927, 832, 1079, 834, 2979, 836, 7990, 838, 13799, 840, 5397, 841, 7709, 842, 7939, 843, 2997, 844, 59099, 845, 2959, 846, 119979, 847, 1097, 848, 159979, 849, 1992, 850, 1727, 852, 2199, 854, 2977, 856, 7790, 858, 11997, 860, 7957, 861, 13599, 862, 59909, 863, 9079, 864, 119997, 865, 7907, 866, 159997, 867, 3099, 868, 197999, 869, 2995, 870, 1392, 872, 1507, 874, 1709, 876, 3909, 878, 9097, 880, 59990, 881, 15979, 882, 135999, 883, 13979, 884, 175999, 885, 13779, 886, 199799, 887, 9709, 888, 399999, 889, 7970, 890, 1525, 892, 1730, 894, 2379, 896, 3079, 897, 936, 898, 9059, 899, 916, 901, 1241, 904, 1112, 905, 1162, 906, 1374, 908, 1549, 909, 1146, 910, 1256, 912, 1065, 914, 958, 921, 1089, 922, 1121, 923, 1078, 924, 1419, 925, 1265, 926, 1552, 928, 1772, 929, 1180, 930, 1038, 932, 1087, 934, 983, 940, 1145, 941, 961, 942, 1437, 943, 1034, 944, 1594, 945, 1098, 946, 1901, 947, 1234, 948, 2397, 949, 1289, 950, 1216, 952, 1298, 954, 1104, 956, 985, 960, 1455, 962, 1699, 963, 978, 964, 1910, 965, 1012, 966, 2559, 967, 1043, 968, 3097, 969, 1122, 972, 1083, 974, 1243, 976, 1067, 980, 1723, 982, 1925, 984, 2577, 986, 3709, 987, 1023, 988, 9077, 989, 1021, 990, 1164, 992, 1258, 994, 1412, 996, 1140, 998, 1045, 1000, 1154, 1001, 1054, 1002, 1473, 1003, 1076, 1004, 1732, 1005, 1212, 1006, 1949, 1007, 1267, 1008, 2595, 1009, 1421, 1010, 1069, 1020, 1491, 1022, 1747, 1024, 1952, 1025, 1096, 1026, 2739, 1027, 1223, 1028, 3790, 1029, 1221, 1030, 1232, 1032, 1047, 1040, 1774, 1042, 1994, 1044, 2757, 1046, 3907, 1048, 9095, 1049, 1102, 1050, 1245, 1052, 1120, 1060, 2159, 1062, 2775, 1064, 3970, 1066, 9299, 1068, 13797, 1070, 1276, 1072, 1247, 1074, 1203, 1080, 2793, 1082, 4999, 1084, 9509, 1086, 13959, 1088, 15997, 1090, 1445, 1092, 1254, 1094, 1201, 1100, 1210, 1111, 1274, 1113, 1269, 1114, 1271, 1115, 1306, 1116, 1479, 1117, 1454, 1118, 1570, 1119, 1236, 1123, 1169, 1131, 1296, 1132, 1196, 1133, 1324, 1134, 1512, 1135, 1469, 1136, 1912, 1137, 1263, 1138, 2177, 1139, 1261, 1141, 1301, 1143, 1521, 1147, 1211, 1148, 1399, 1151, 1342, 1153, 1496, 1155, 1278, 1156, 1310, 1157, 1285, 1158, 1497, 1161, 1503, 1163, 1921, 1165, 1325, 1166, 1579, 1167, 1545, 1168, 1799, 1171, 1508, 1173, 1287, 1174, 1334, 1175, 1414, 1176, 1554, 1178, 1945, 1179, 1182, 1181, 1705, 1183, 2195, 1184, 1597, 1185, 1527, 1186, 1979, 1187, 1954, 1188, 3999, 1189, 1349, 1191, 1308, 1193, 1441, 1197, 1218, 1198, 1352, 1200, 1569, 1202, 1969, 1204, 2339, 1205, 1225, 1206, 2919, 1207, 1304, 1208, 5059, 1209, 1401, 1213, 1343, 1220, 1996, 1222, 2357, 1224, 2937, 1226, 5077, 1227, 1230, 1228, 9590, 1229, 1249, 1231, 1367, 1238, 1555, 1240, 2375, 1242, 2955, 1244, 5095, 1246, 9707, 1248, 13977, 1250, 1252, 1260, 2973, 1262, 5299, 1264, 9770, 1266, 13995, 1268, 17599, 1270, 1322, 1272, 1302, 1280, 5509, 1281, 1359, 1282, 9905, 1283, 1573, 1284, 15399, 1286, 17779, 1288, 17999, 1290, 1410, 1292, 1294, 1300, 1340, 1311, 1425, 1312, 1376, 1313, 1360, 1314, 1596, 1315, 1526, 1316, 2119, 1317, 1326, 1318, 2393, 1319, 1456, 1320, 1407, 1321, 1415, 1328, 1591, 1331, 1423, 1333, 1544, 1335, 1344, 1336, 1394, 1337, 1465, 1338, 1530, 1341, 1659, 1345, 1433, 1346, 1717, 1348, 3377, 1351, 1562, 1353, 1362, 1354, 1451, 1355, 1489, 1356, 1677, 1358, 2137, 1361, 2155, 1363, 1439, 1364, 1735, 1365, 1695, 1366, 3395, 1368, 5559, 1369, 1514, 1371, 1380, 1373, 1498, 1377, 1416, 1378, 1457, 1381, 2519, 1382, 1753, 1383, 1572, 1384, 3539, 1385, 2173, 1386, 5577, 1387, 1475, 1388, 7975, 1389, 1734, 1391, 1528, 1395, 1458, 1396, 1541, 1398, 1743, 1400, 2191, 1402, 2537, 1403, 1702, 1404, 2991, 1405, 1409, 1406, 5590, 1408, 9929, 1411, 1493, 1413, 1767, 1417, 1556, 1418, 1759, 1420, 2555, 1422, 3039, 1424, 5707, 1426, 9950, 1427, 1720, 1428, 15579, 1429, 1490, 1430, 2017, 1431, 1776, 1432, 1511, 1434, 1515, 1435, 1565, 1436, 1771, 1438, 3557, 1440, 3057, 1442, 5770, 1443, 1533, 1444, 9992, 1446, 15597, 1447, 1700, 1448, 17797, 1449, 1500, 1450, 1502, 1452, 1902, 1453, 1589, 1460, 5905, 1461, 1539, 1462, 11399, 1463, 1915, 1464, 15759, 1466, 17959, 1467, 1551, 1468, 19799, 1470, 1704, 1471, 1598, 1472, 2035, 1474, 2039, 1476, 1713, 1478, 1933, 1480, 11579, 1481, 1777, 1482, 15777, 1483, 3575, 1484, 17977, 1485, 1557, 1486, 19979, 1487, 1951, 1488, 39999, 1492, 1520, 1494, 1920, 1504, 1904, 1506, 2019, 1513, 1580, 1516, 1523, 1517, 1546, 1518, 1701, 1522, 1922, 1524, 2037, 1531, 1601, 1532, 1558, 1534, 1619, 1535, 1564, 1536, 1899, 1538, 2317, 1540, 1940, 1542, 2055, 1543, 1637, 1547, 1561, 1548, 1575, 1553, 1582, 1560, 2073, 1563, 1914, 1566, 1593, 1567, 1655, 1568, 1795, 1571, 1612, 1574, 1585, 1576, 1673, 1578, 1941, 1581, 1710, 1583, 2335, 1584, 1719, 1586, 1939, 1587, 1956, 1588, 1997, 1600, 2573, 1602, 3075, 1603, 2057, 1604, 5929, 1605, 2091, 1606, 11597, 1607, 2053, 1608, 15795, 1609, 2015, 1610, 1711, 1611, 1725, 1613, 2353, 1614, 1737, 1615, 1679, 1616, 1957, 1617, 1965, 1618, 3599, 1620, 3093, 1621, 1715, 1622, 5950, 1623, 1731, 1624, 11759, 1625, 1900, 1626, 15939, 1627, 2075, 1628, 17995, 1629, 2109, 1630, 2093, 1631, 2371, 1632, 1911, 1633, 1697, 1634, 3139, 1635, 1989, 1636, 3593, 1638, 5595, 1639, 1691, 1640, 5992, 1641, 1755, 1642, 11777, 1643, 3157, 1644, 15957, 1645, 1733, 1646, 19399, 1647, 2079, 1648, 19997, 1649, 2059, 1650, 2190, 1651, 1712, 1652, 2077, 1653, 1998, 1654, 1751] 

Somme des pairs [0, 2, 2, 4, 6, 6, 6, 8, 12, 12, 12, 16, 22, 22, 22, 26, 34, 34, 34, 40, 40, 42, 42, 44, 44, 48, 54, 54, 62, 62, 62, 68, 70, 70, 72, 74, 78, 78, 84, 84, 86, 90, 98, 98, 108, 108, 110, 114, 114, 118, 118, 122, 122, 130, 138, 138, 142, 142, 146, 146, 152, 152, 160, 160, 170, 170, 182, 182, 186, 190, 190, 196, 196, 202, 204, 206, 210, 212, 212, 218, 224, 224, 232, 232, 232, 240, 246, 246, 252, 252, 260, 260, 266, 266, 276, 276, 282, 282, 294, 294, 308, 308, 314, 314, 314, 320, 320, 328, 334, 336, 336, 344, 352, 352, 360, 360, 368, 368, 378, 378, 386, 386, 398, 398, 406, 406, 420, 420, 428, 428, 444, 444, 452, 452, 452, 460, 464, 468, 468, 476, 482, 484, 492, 492, 492, 502, 502, 504, 504, 506, 508, 510, 510, 514, 518, 518, 518, 524, 530, 530, 530, 536, 544, 544, 544, 552, 552, 556, 556, 560, 562, 562, 566, 566, 572, 572, 574, 578, 586, 586, 588, 592, 602, 602, 604, 612, 612, 618, 618, 622, 624, 628, 636, 638, 642, 642, 648, 648, 656, 656, 666, 666, 678, 678, 682, 688, 688, 696, 698, 704, 704, 710, 716, 716, 724, 724, 734, 734, 746, 746, 760, 760, 760, 768, 768, 774, 776, 782, 786, 788, 788, 796, 796, 806, 814, 814, 824, 824, 832, 834, 846, 846, 860, 860, 868, 870, 886, 886, 886, 894, 896, 902, 902, 910, 914, 916, 916, 926, 928, 930, 932, 934, 938, 940, 946, 946, 948, 952, 960, 960, 970, 970, 972, 978, 980, 982, 986, 986, 992, 992, 994, 998, 1006, 1006, 1016, 1016, 1018, 1024, 1028, 1030, 1034, 1034, 1040, 1040, 1048, 1048, 1052, 1052, 1062, 1062, 1074, 1074, 1078, 1080, 1082, 1084, 1086, 1090, 1092, 1096, 1098, 1106, 1112, 1112, 1118, 1118, 1126, 1126, 1136, 1136, 1142, 1142, 1154, 1154, 1168, 1168, 1174, 1174, 1176, 1178, 1180, 1186, 1190, 1190, 1192, 1196, 1202, 1202, 1204, 1212, 1220, 1220, 1230, 1230, 1232, 1240, 1248, 1248, 1256, 1256, 1266, 1266, 1278, 1278, 1286, 1286, 1300, 1300, 1308, 1308, 1324, 1324, 1332, 1334, 1336, 1342, 1344, 1348, 1354, 1356, 1364, 1364, 1366, 1372, 1382, 1382, 1392, 1392, 1402, 1402, 1414, 1414, 1428, 1428, 1438, 1438, 1454, 1454, 1464, 1464, 1482, 1482, 1492, 1492, 1494, 1500, 1504, 1508, 1510, 1516, 1522, 1524, 1526, 1534, 1542, 1542, 1552, 1552, 1554, 1562, 1562, 1568, 1568, 1576, 1580, 1580, 1580, 1588, 1594, 1594, 1594, 1602, 1610, 1610, 1610, 1618, 1626, 1626, 1628, 1632, 1636, 1636, 1638, 1644, 1650, 1650, 1652, 1660, 1668, 1668, 1670, 1676, 1686, 1686, 1688, 1694, 1694, 1702, 1704, 1708, 1712, 1714, 1720, 1720, 1728, 1728, 1738, 1738, 1742, 1746, 1758, 1758, 1762, 1766, 1766, 1774, 1776, 1780, 1784, 1790, 1796, 1796, 1804, 1804, 1814, 1814, 1826, 1826, 1840, 1840, 1840, 1848, 1850, 1856, 1860, 1864, 1872, 1872, 1880, 1880, 1890, 1890, 1902, 1902, 1916, 1916, 1932, 1932, 1934, 1938, 1942, 1948, 1954, 1958, 1962, 1964, 1970, 1972, 1976, 1978, 1986, 1986, 1990, 1994, 2004, 2004, 2016, 2016, 2020, 2024, 2028, 2030, 2036, 2036, 2044, 2044, 2054, 2054, 2066, 2066, 2070, 2074, 2080, 2080, 2086, 2086, 2094, 2094, 2100, 2100, 2110, 2110, 2116, 2118, 2130, 2130, 2136, 2136, 2150, 2150, 2156, 2158, 2162, 2162, 2168, 2168, 2176, 2176, 2180, 2184, 2194, 2194, 2198, 2202, 2214, 2214, 2218, 2224, 2232, 2232, 2240, 2240, 2250, 2250, 2258, 2258, 2270, 2270, 2278, 2278, 2292, 2292, 2300, 2300, 2316, 2316, 2324, 2326, 2330, 2334, 2340, 2342, 2350, 2350, 2354, 2360, 2370, 2370, 2374, 2382, 2394, 2394, 2398, 2404, 2414, 2414, 2424, 2424, 2436, 2436, 2446, 2446, 2460, 2460, 2470, 2470, 2486, 2486, 2496, 2496, 2514, 2514, 2524, 2524, 2528, 2532, 2538, 2538, 2542, 2548, 2556, 2556, 2560, 2568, 2578, 2578, 2582, 2588, 2600, 2600, 2604, 2612, 2624, 2624, 2636, 2636, 2650, 2650, 2662, 2662, 2678, 2678, 2690, 2690, 2708, 2708, 2720, 2720, 2740, 2740, 2752, 2752, 2756, 2760, 2764, 2772, 2778, 2780, 2784, 2790, 2798, 2800, 2804, 2812, 2822, 2822, 2826, 2834, 2846, 2846, 2850, 2856, 2856, 2864, 2866, 2870, 2870, 2876, 2880, 2882, 2882, 2890, 2896, 2896, 2904, 2904, 2904, 2912, 2912, 2920, 2922, 2930, 2932, 2936, 2938, 2944, 2948, 2950, 2952, 2958, 2964, 2966, 2968, 2976, 2984, 2984, 2986, 2992, 3002, 3002, 3004, 3010, 3010, 3018, 3020, 3026, 3030, 3032, 3038, 3038, 3042, 3048, 3056, 3058, 3062, 3068, 3078, 3078, 3082, 3088, 3100, 3100, 3104, 3112, 3112, 3120, 3122, 3130, 3134, 3140, 3146, 3148, 3156, 3158, 3168, 3168, 3180, 3180, 3194, 3194, 3200, 3206, 3208, 3216, 3220, 3226, 3232, 3238, 3246, 3248, 3258, 3258, 3270, 3270, 3284, 3286, 3302, 3302, 3302, 3310, 3312, 3318, 3322, 3330, 3336, 3342, 3348, 3350, 3356, 3356, 3364, 3364, 3370, 3370, 3380, 3380, 3386, 3390, 3402, 3402, 3408, 3410, 3424, 3424, 3430, 3432, 3438, 3438, 3446, 3446, 3456, 3456, 3468, 3468, 3482, 3482, 3490, 3490, 3498, 3500, 3510, 3510, 3518, 3518, 3530, 3530, 3538, 3542, 3556, 3556, 3564, 3564, 3580, 3580, 3588, 3592, 3598, 3598, 3606, 3606, 3616, 3616, 3628, 3628, 3642, 3642, 3652, 3652, 3662, 3662, 3674, 3674, 3684, 3684, 3698, 3698, 3708, 3708, 3724, 3724, 3734, 3734, 3752, 3752, 3762, 3762, 3768, 3772, 3780, 3782, 3792, 3792, 3804, 3804, 3810, 3816, 3830, 3830, 3836, 3842, 3854, 3854, 3866, 3866, 3880, 3880, 3892, 3892, 3908, 3908, 3920, 3920, 3938, 3938, 3950, 3950, 3970, 3970, 3982, 3982, 3988, 3988, 3996, 3998, 4008, 4008, 4020, 4020, 4026, 4032, 4046, 4046, 4052, 4060, 4074, 4074, 4088, 4088, 4104, 4104, 4118, 4118, 4136, 4136, 4150, 4150, 4170, 4170, 4184, 4184, 4206, 4206, 4220, 4220, 4226, 4230, 4238, 4238, 4244, 4248, 4258, 4258, 4264, 4272, 4284, 4284, 4290, 4296, 4310, 4312, 4318, 4326, 4326, 4334, 4334, 4340, 4346, 4348, 4348, 4356, 4364, 4366, 4366, 4374, 4376, 4384, 4386, 4394, 4396, 4404, 4410, 4412, 4414, 4422, 4430, 4432, 4434, 4440, 4450, 4450, 4450, 4456, 4458, 4462, 4466, 4474, 4478, 4484, 4490, 4492, 4496, 4504, 4512, 4514, 4518, 4526, 4536, 4536, 4540, 4544, 4556, 4556, 4560, 4568, 4570, 4576, 4580, 4588, 4594, 4596, 4604, 4604, 4614, 4614, 4620, 4628, 4640, 4640, 4646, 4654, 4668, 4668, 4674, 4682, 4682, 4690, 4692, 4698, 4702, 4708, 4714, 4722, 4730, 4732, 4742, 4744, 4756, 4756, 4770, 4772, 4788, 4788, 4792, 4798, 4804, 4812, 4820, 4828, 4836, 4836, 4844, 4844, 4854, 4854, 4862, 4864, 4876, 4876, 4884, 4886, 4900, 4900, 4908, 4908, 4924, 4926, 4934, 4934, 4942, 4944, 4954, 4956, 4964, 4964, 4976, 4976, 4990, 4990, 5006, 5006, 5016, 5016, 5026, 5028, 5040, 5040, 5050, 5050, 5064, 5064, 5074, 5074, 5090, 5090, 5100, 5100, 5118, 5118, 5128, 5128, 5136, 5136, 5144, 5146, 5156, 5156, 5168, 5170, 5184, 5184, 5200, 5200, 5212, 5212, 5224, 5224, 5238, 5238, 5250, 5252, 5268, 5268, 5280, 5282, 5300, 5300, 5312, 5312, 5332, 5332, 5344, 5346, 5354, 5356, 5366, 5368, 5380, 5382, 5396, 5396, 5412, 5412, 5426, 5426, 5440, 5440, 5456, 5456, 5470, 5470, 5488, 5488, 5502, 5502, 5522, 5522, 5536, 5536, 5558, 5558, 5572, 5574, 5582, 5584, 5594, 5594, 5606, 5606, 5620, 5620, 5636, 5636, 5652, 5652, 5668, 5668, 5686, 5686, 5702, 5702, 5722, 5722, 5738, 5738, 5760, 5760, 5776, 5776, 5800, 5800, 5816, 5816, 5824, 5826, 5836, 5836, 5848, 5850, 5864, 5864, 5872, 5878, 5894, 5894, 5902, 5908, 5908, 5914, 5918, 5920, 5920, 5928, 5934, 5938, 5946, 5950, 5950, 5960, 5960, 5968, 5970, 5976, 5980, 5988, 5990, 5998, 6002, 6004, 6006, 6014, 6020, 6024, 6026, 6034, 6042, 6044, 6054, 6056, 6058, 6066, 6066, 6074, 6076, 6084, 6088, 6096, 6100, 6104, 6108, 6114, 6120, 6124, 6128, 6132, 6140, 6144, 6148, 6156, 6166, 6166, 6170, 6176, 6188, 6190, 6194, 6204, 6204, 6212, 6214, 6224, 6228, 6232, 6238, 6246, 6252, 6256, 6264, 6270, 6276, 6284, 6294, 6294, 6300, 6302, 6314, 6316, 6322, 6326, 6340, 6340, 6346, 6350, 6352, 6360, 6364, 6370, 6376, 6382, 6390, 6392, 6402, 6404, 6416, 6418, 6432, 6432, 6440, 6442, 6458, 6458, 6466, 6468, 6468, 6478, 6480, 6490, 6494, 6500, 6506, 6510, 6518, 6522, 6522, 6526, 6526, 6530, 6532, 6536, 6536, 6542, 6546, 6548, 6548, 6552, 6558, 6562, 6562, 6570, 6578, 6580, 6580, 6586, 6586, 6592, 6594, 6598, 6602, 6606, 6612, 6614, 6616, 6622, 6630, 6632, 6634, 6638, 6648, 6648, 6650, 6654, 6654, 6658, 6660, 6664, 6668, 6672, 6678, 6682, 6690, 6692, 6702, 6702, 6714, 6714, 6718, 6720, 6720, 6726, 6728, 6730, 6736, 6738, 6746, 6748, 6758, 6758, 6770, 6772, 6786, 6786, 6786, 6794, 6796, 6802, 6806, 6808, 6816, 6818, 6828, 6832, 6844, 6844, 6858, 6858, 6874, 6874, 6874, 6882, 6884, 6890, 6894, 6896, 6896, 6898, 6898, 6904, 6904, 6912, 6916, 6918, 6918, 6924, 6930, 6934, 6934, 6942, 6950, 6950, 6950, 6958, 6960, 6966, 6966, 6974, 6976, 6982, 6982, 6988, 6992, 6994, 6994, 7004, 7010, 7012, 7012, 7020, 7028, 7030, 7030, 7038, 7042, 7042, 7046, 7048, 7052, 7054, 7066, 7066, 7066, 7072, 7072, 7082, 7082, 7092, 7098, 7098, 7098, 7108, 7116, 7120, 7126, 7126, 7132, 7134, 7140, 7142, 7154, 7154, 7160, 7164, 7178, 7178, 7178, 7186, 7186, 7196, 7200, 7204, 7204, 7212, 7218, 7222, 7230, 7234, 7234, 7244, 7252, 7252, 7260, 7262, 7274, 7274, 7282, 7284, 7298, 7298, 7306, 7310, 7326, 7326, 7334, 7338, 7338, 7346, 7346, 7354, 7354, 7364, 7372, 7374, 7376, 7382, 7386, 7392, 7398, 7400, 7402, 7406, 7414, 7416, 7418, 7422, 7432, 7432, 7434, 7438, 7440, 7444, 7448, 7454, 7460, 7462, 7470, 7472, 7482, 7482, 7486, 7488, 7500, 7500, 7504, 7510, 7512, 7518, 7528, 7528, 7534, 7536, 7544, 7546, 7556, 7556, 7568, 7568, 7582, 7582, 7584, 7588, 7596, 7598, 7608, 7610, 7622, 7622, 7636, 7636, 7652, 7652, 7654, 7658, 7662, 7664, 7674, 7674, 7684, 7684, 7696, 7696, 7706, 7706, 7720, 7720, 7736, 7736, 7754, 7754, 7756, 7760, 7764, 7770, 7770, 7774, 7774, 7780, 7782, 7788, 7788, 7794, 7798, 7804, 7804, 7812, 7818, 7820, 7820, 7828, 7836, 7838, 7838, 7848, 7850, 7854, 7856, 7860, 7870, 7870, 7870, 7876, 7876, 7884, 7884, 7892, 7898, 7902, 7902, 7912, 7920, 7920, 7924, 7930, 7934, 7938, 7948, 7948, 7960, 7960, 7960, 7968, 7968, 7976, 7980, 7984, 7984, 7996, 8002, 8008, 8016, 8018, 8024, 8026, 8032, 8036, 8046, 8046, 8052, 8058, 8070, 8070, 8084, 8084, 8090, 8094, 8094, 8102, 8102, 8114, 8114, 8124, 8132, 8136, 8144, 8146, 8156, 8156, 8164, 8166, 8178, 8178, 8186, 8188, 8202, 8202, 8210, 8214, 8230, 8230, 8238, 8242, 8242, 8252, 8252, 8264, 8270, 8274, 8282, 8286, 8290, 8292, 8298, 8300, 8304, 8306, 8314, 8316, 8320, 8324, 8334, 8334, 8346, 8348, 8352, 8356, 8360, 8366, 8370, 8376, 8388, 8388, 8394, 8396, 8404, 8404, 8414, 8414, 8426, 8426, 8432, 8434, 8448, 8448, 8454, 8458, 8462, 8464, 8468, 8474, 8480, 8480, 8488, 8488, 8492, 8498, 8508, 8508, 8520, 8520, 8528, 8528, 8538, 8538, 8546, 8546, 8558, 8560, 8574, 8574, 8582, 8582, 8598, 8598, 8606, 8606, 8610, 8612, 8618, 8620, 8624, 8632, 8642, 8642, 8652, 8652, 8664, 8664, 8674, 8674, 8688, 8688, 8704, 8704, 8714, 8714, 8732, 8732, 8736, 8740, 8744, 8752, 8758, 8760, 8768, 8770, 8780, 8780, 8792, 8792, 8804, 8804, 8816, 8816, 8830, 8830, 8842, 8842, 8858, 8858, 8870, 8870, 8888, 8888, 8900, 8900, 8920, 8920, 8926, 8928, 8936, 8938, 8942, 8946, 8952, 8954, 8954, 8962, 8968, 8970, 8970, 8980, 8988, 8988, 8992, 8996, 9002, 9004, 9004, 9010, 9012, 9020, 9024, 9030, 9030, 9040, 9046, 9054, 9062, 9064, 9068, 9072, 9078, 9080, 9084, 9090, 9094, 9100, 9112, 9112, 9112, 9122, 9128, 9130, 9136, 9140, 9152, 9152, 9158, 9164, 9178, 9178, 9178, 9186, 9190, 9198, 9204, 9210, 9218, 9222, 9230, 9230, 9238, 9240, 9252, 9252, 9266, 9266, 9274, 9280, 9296, 9296, 9302, 9304, 9312, 9312, 9318, 9320, 9330, 9332, 9338, 9340, 9352, 9352, 9358, 9360, 9374, 9374, 9380, 9382, 9388, 9388, 9394, 9396, 9402, 9404, 9414, 9414, 9420, 9426, 9438, 9438, 9444, 9450, 9464, 9464, 9472, 9472, 9480, 9480, 9490, 9490, 9498, 9498, 9510, 9510, 9518, 9518, 9532, 9532, 9540, 9542, 9558, 9558, 9566, 9568, 9574, 9576, 9582, 9584, 9592, 9592, 9598, 9604, 9614, 9614, 9620, 9628, 9640, 9640, 9654, 9654, 9660, 9666, 9676, 9678, 9688, 9688, 9700, 9700, 9710, 9710, 9724, 9724, 9734, 9734, 9750, 9750, 9760, 9762, 9780, 9780, 9790, 9792, 9798, 9800, 9806, 9808, 9816, 9818, 9824, 9832, 9842, 9842]

Somme des impairs [0, 4, 5, 8, 8, 12, 15, 16, 16, 24, 29, 32, 32, 44, 51, 52, 52, 68, 77, 80, 81, 84, 86, 88, 94, 96, 97, 108, 109, 124, 134, 136, 136, 140, 141, 148, 148, 156, 156, 168, 173, 180, 180, 196, 196, 216, 225, 228, 231, 236, 242, 244, 254, 260, 263, 276, 276, 284, 285, 292, 292, 304, 304, 320, 320, 340, 340, 364, 373, 380, 385, 392, 398, 404, 409, 412, 417, 424, 434, 436, 441, 448, 453, 464, 478, 480, 480, 492, 493, 504, 504, 520, 523, 532, 532, 552, 557, 564, 564, 588, 588, 616, 625, 628, 636, 640, 650, 656, 663, 672, 686, 688, 695, 704, 704, 720, 721, 736, 736, 756, 759, 772, 772, 796, 801, 812, 812, 840, 847, 856, 856, 888, 897, 904, 914, 920, 929, 936, 950, 952, 961, 968, 977, 984, 1002, 1004, 1005, 1008, 1010, 1012, 1013, 1020, 1024, 1028, 1029, 1036, 1042, 1048, 1049, 1060, 1068, 1072, 1073, 1088, 1098, 1104, 1106, 1112, 1117, 1120, 1121, 1124, 1125, 1132, 1133, 1144, 1150, 1156, 1157, 1172, 1180, 1184, 1185, 1204, 1214, 1224, 1228, 1236, 1241, 1244, 1248, 1256, 1260, 1276, 1277, 1284, 1285, 1296, 1297, 1312, 1313, 1332, 1333, 1356, 1366, 1376, 1382, 1392, 1398, 1408, 1417, 1420, 1421, 1432, 1433, 1448, 1449, 1468, 1469, 1492, 1493, 1520, 1528, 1536, 1545, 1548, 1556, 1564, 1572, 1576, 1589, 1592, 1609, 1612, 1613, 1628, 1629, 1648, 1652, 1668, 1669, 1692, 1693, 1720, 1728, 1740, 1741, 1772, 1782, 1788, 1798, 1804, 1817, 1820, 1830, 1832, 1849, 1852, 1852, 1860, 1861, 1868, 1868, 1880, 1880, 1892, 1897, 1904, 1904, 1920, 1920, 1940, 1949, 1956, 1957, 1964, 1965, 1972, 1973, 1984, 1990, 1996, 1997, 2012, 2013, 2032, 2042, 2048, 2048, 2060, 2061, 2068, 2068, 2080, 2080, 2096, 2101, 2104, 2104, 2124, 2124, 2148, 2157, 2160, 2164, 2168, 2174, 2180, 2188, 2192, 2202, 2212, 2212, 2224, 2225, 2236, 2236, 2252, 2252, 2272, 2277, 2284, 2284, 2308, 2308, 2336, 2345, 2348, 2353, 2356, 2362, 2372, 2377, 2380, 2388, 2392, 2397, 2404, 2414, 2424, 2429, 2440, 2445, 2460, 2474, 2480, 2480, 2496, 2497, 2512, 2512, 2532, 2532, 2556, 2561, 2572, 2572, 2600, 2607, 2616, 2616, 2648, 2657, 2668, 2676, 2684, 2694, 2696, 2703, 2712, 2719, 2728, 2742, 2744, 2751, 2764, 2764, 2784, 2785, 2804, 2804, 2828, 2828, 2856, 2861, 2876, 2876, 2908, 2915, 2928, 2928, 2964, 2973, 2984, 2994, 3000, 3009, 3016, 3028, 3032, 3041, 3048, 3062, 3068, 3077, 3084, 3093, 3104, 3122, 3124, 3128, 3136, 3142, 3152, 3155, 3160, 3168, 3176, 3179, 3188, 3198, 3204, 3207, 3220, 3224, 3236, 3240, 3252, 3256, 3264, 3267, 3272, 3278, 3288, 3291, 3300, 3308, 3320, 3323, 3336, 3346, 3352, 3355, 3372, 3384, 3388, 3394, 3404, 3410, 3416, 3419, 3428, 3431, 3440, 3443, 3456, 3459, 3476, 3486, 3492, 3495, 3516, 3528, 3532, 3540, 3548, 3556, 3560, 3568, 3580, 3583, 3592, 3595, 3608, 3611, 3628, 3631, 3652, 3655, 3680, 3690, 3696, 3706, 3712, 3722, 3728, 3731, 3744, 3748, 3760, 3763, 3780, 3783, 3804, 3807, 3832, 3835, 3864, 3876, 3876, 3888, 3896, 3908, 3916, 3917, 3928, 3928, 3944, 3947, 3956, 3956, 3972, 3977, 3988, 3988, 4008, 4008, 4032, 4041, 4048, 4049, 4060, 4061, 4072, 4073, 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