Ajouter
à n le 2e nombre qui ne
divise pas n.
Itérer.
Le message suivant fut envoyé par
l’auteur aux listes MathFun
et SeqFan,
fin juin 2008
(traduction
juste après -- je crois que j’ai confondu divider et divisor) :
__________
Hello MathFun & SeqFans,
Rule:
< Add to n its
second smallest non-divider.
Loop. >
Let's start with n = 7, for instance
Is 1 a divider of 7? yes
2 no
3 no --> then new n = 7+3 = 10
Is 1 a divider of 10? yes
2 yes
3 no
4 no --> then new n =10+4= 14
Is 1 a divider of 14? yes
2 yes
3 no
4 no --> then new n =14+4= 18
Is 1 a divider of 18? yes
2 yes
3 yes
4 no
5 no --> then new n =18+5= 23
... etc.
Sequence starting with 7 is: 7,10,14,18,23,...
Sequence starting with 1 is: 1,4,9,13,16,21,... [not in the OEIS]
Sequence starting with 2 is: 2,6,11,14,18,23,... [merges with "7-seq"]
Sequence starting with 3 is: 3,7,10,14,... [merges with "7-seq"]
Sequence starting with 5 is: 5,8,13,16,... [merges with "1-seq"]
etc.
We might map those sequences
like this:
1--4--9---13--16--21--25--28--33--37--40 ...
| |
5--8---+ 29--32--+ 41 ...
|
2--6--11--14--18--23--26--30------+
| |
3--7--10--+ 17--20--+ 36--43 ...
| |
12--19--22--+ 39--+
|
15--+ 24--31--34--38--42 ...
| |
27--+ 35--+
What number starts the longest sequence containing 2008?
Best,
É.
---
P.-S.
This rule is
not very productive: < Add to
n its first smallest non-divider. Loop. >
What about: < Add to n its third
smallest non-divider. Loop. >
Etc.
__________
On a compris, il s’agit d’ajouter à un nombre n le deuxième nombre qui ne divise pas n, puis d’itérer.
(C’est la loi dite « D2 »
selon la terminologie adoptée par Jacques
Tramu, lequel a travaillé sur ces suites.)
Prenons 7 par exemple :
1 divise-t-il 7 ? oui
2 non
3 non --> alors le nouveau n
= 7+3 = 10
1 divise-t-il 10 ? oui
2 oui
3 non
4 non --> alors le nouveau n
=10+4= 14
1 divise-t-il 14 ? oui
2 oui
3 non
4 non --> alors le nouveau n
=14+4= 18
1 divise-t-il 18 ? oui
2 oui
3 oui
4 non
5 non --> alors le nouveau n
=18+5= 23
... etc.
La suite commençant par 7 [baptisons-la D2(7)] est donc :
D2(7) = 7,
10, 14, 18, 23, 26, 30, 37, 40, 46, 50...
La suite commençant par 1, [ou D2(1)], est :
D2(1) = 1, 4, 9, 13, 16, 21, 25, 28, 33, 37, 40,
46, 50...
Les termes soulignés dans D2(1) et D2(7) sont identiques à partir d’un certain point : on dit de
ces deux suites qu’elles confluent en 37.
Voici 500 termes itérés de D2(1) :
D2(1) = 1, 4, 9, 13, 16, 21, 25, 28, 33, 37, 40, 46, 50, 54,
59, 62, 66, 71, 74, 78, 83, 86, 90, 97, 100, 106, 110, 114, 119, 122, 126, 131,
134, 138, 143, 146, 150, 157, 160, 166, 170, 174, 179, 182, 186, 191, 194, 198,
203, 206, 210, 218, 222, 227, 230, 234, 239, 242, 246, 251, 254, 258, 263, 266,
270, 277, 280, 286, 290, 294, 299, 302, 306, 311, 314, 318, 323, 326, 330, 337,
340, 346, 350, 354, 359, 362, 366, 371, 374, 378, 383, 386, 390, 397, 400, 406,
410, 414, 419, 422, 426, 431, 434, 438, 443, 446, 450, 457, 460, 466, 470, 474,
479, 482, 486, 491, 494, 498, 503, 506, 510, 517, 520, 526, 530, 534, 539, 542,
546, 551, 554, 558, 563, 566, 570, 577, 580, 586, 590, 594, 599, 602, 606, 611,
614, 618, 623, 626, 630, 638, 642, 647, 650, 654, 659, 662, 666, 671, 674, 678, 683, 686, 690, 697,
700, 706, 710, 714, 719, 722, 726, 731, 734, 738, 743, 746, 750, 757, 760, 766,
770, 774, 779, 782, 786, 791, 794, 798, 803, 806, 810, 817, 820, 826, 830, 834,
839, 842, 846, 851, 854, 858, 863, 866, 870, 877, 880, 886, 890, 894, 899, 902,
906, 911, 914, 918, 923, 926, 930, 937, 940, 946, 950, 954, 959, 962, 966, 971,
974, 978, 983, 986, 990, 997, 1000, 1006, 1010, 1014, 1019, 1022, 1026, 1031,
1034, 1038, 1043, 1046, 1050, 1058, 1062, 1067, 1070, 1074, 1079, 1082, 1086,
1091, 1094, 1098, 1103, 1106, 1110, 1117, 1120, 1126, 1130, 1134, 1139, 1142,
1146, 1151, 1154, 1158, 1163, 1166, 1170, 1177, 1180, 1186, 1190, 1194, 1199,
1202, 1206, 1211, 1214, 1218, 1223, 1226, 1230, 1237, 1240, 1246, 1250, 1254,
1259, 1262, 1266, 1271, 1274, 1278, 1283, 1286, 1290, 1297, 1300, 1306, 1310,
1314, 1319, 1322, 1326, 1331, 1334, 1338, 1343, 1346, 1350, 1357, 1360, 1366,
1370, 1374, 1379, 1382, 1386, 1391, 1394, 1398, 1403, 1406, 1410, 1417, 1420,
1426, 1430, 1434, 1439, 1442, 1446, 1451, 1454, 1458, 1463, 1466, 1470, 1478,
1482, 1487, 1490, 1494, 1499, 1502, 1506, 1511, 1514, 1518, 1523, 1526, 1530,
1537, 1540, 1546, 1550, 1554, 1559, 1562, 1566, 1571, 1574, 1578, 1583, 1586,
1590, 1597, 1600, 1606, 1610, 1614, 1619, 1622, 1626, 1631, 1634, 1638, 1643,
1646, 1650, 1657, 1660, 1666, 1670, 1674, 1679, 1682, 1686, 1691, 1694, 1698,
1703, 1706, 1710, 1717, 1720, 1726, 1730, 1734, 1739, 1742, 1746, 1751, 1754,
1758, 1763, 1766, 1770, 1777, 1780, 1786, 1790, 1794, 1799, 1802, 1806, 1811,
1814, 1818, 1823, 1826, 1830, 1837, 1840, 1846, 1850, 1854, 1859, 1862, 1866,
1871, 1874, 1878, 1883, 1886, 1890, 1898, 1902, 1907, 1910, 1914, 1919, 1922,
1926, 1931, 1934, 1938, 1943, 1946, 1950, 1957, 1960, 1966, 1970, 1974, 1979,
1982, 1986, 1991, 1994, 1998, 2003, 2006, 2010, 2017, 2020, 2026, 2030, 2034,
2039, 2042, 2046, 2051, 2054, 2058, 2063, 2066, 2070, 2077, 2080, 2086, 2090,
2094, 2099, 2102, 2106, 2111, 2114, 2118, 2123, 2126, 2130, 2137, 2140, 2146,
2150, 2154, 2159, 2162, 2166, 2171, 2174, 2178, 2183, 2186, 2190, 2197,...
On a vu que certaines suites finissent par mettre leurs pas dans
ceux d’une suite existant déjà. Voici le tableau des suites générées par des
nombres ne figurant pas plus haut dans le tableau lui-même (l’astérisque
indique le point de confluence d’un début de suite avec une suite existant déjà
— merci encore à Jacques T.) :
D2(1) = 1, 4, 9, 13, 16, 21, 25, 28, 33, 37, 40, 46, 50, 54,
59, 62, 66, 71, 74, 78, 83, 86, 90, 97, 100, ...
D2(2) = 2, 6, 11, 14, 18, 23, 26, 30, 37*
D2(3) = 3, 7, 10, 14*
D2(5) = 5, 8, 13*
D2(12) = 12, 19, 22, 26*
D2(15) = 15, 19*
D2(17) = 17, 20, 26*
D2(24) = 24, 31, 34, 38, 42, 47, 50*
D2(27) = 27, 31*
D2(29) = 29, 32, 37*
D2(35) = 35, 38*
D2(36) = 36, 43, 46*
D2(39) = 39, 43*
D2(41) = 41, 44, 49, 52, 57, 61, 64, 69, 73, 76, 81, 85, 88,
93, 97*
D2(45) = 45, 49*
D2(48) = 48, 55, 58, 62*
D2(51) = 51, 55*
D2(53) = 53, 56, 61*
D2(60) = 60, 68, 73*
D2(63) = 63, 67, 70, 74*
D2(65) = 65, 68*
D2(72) = 72, 79, 82, 86*
D2(75) = 75, 79*
D2(77) = 77, 80, 86*
D2(84) = 84, 92, 97*
D2(87) = 87, 91, 94, 98, 102, 107, 110*
D2(89) = 89, 92*
D2(95) = 95, 98*
D2(96) = 96, 103, 106*
D2(99) = 99, 103*
D2(101) = 101, 104, 109, 112, 117, 121, 124, 129, 133, 136, 141,
145, 148, 153, 157*
D2(105) = 105, 109*
D2(108) = 108, 115, 118, 122*
D2(111) = 111, 115*
D2(113) = 113, 116, 121*
D2(120) = 120, 129*
D2(123) = 123, 127, 130, 134*
D2(125) = 125, 128, 133*
D2(132) = 132, 139, 142, 146*
D2(135) = 135,
139*
D2(137) = 137, 140, 146*
D2(144) = 144, 151, 154, 158, 162, 167, 170*
D2(147) = 147, 151*
D2(149) = 149, 152, 157*
D2(155) = 155, 158*
D2(156) = 156, 163, 166*
D2(159) = 159, 163*
D2(161) = 161, 164, 169, 172, 177, 181, 184, 189, 193, 196, 201,
205, 208, 213, 217, 220, 226, 230*
D2(165) = 165, 169*
D2(168) = 168, 177*
D2(171) = 171, 175, 178, 182*
D2(173) = 173, 176, 181*
D2(180) = 180, 188, 193*
D2(183) = 183, 187, 190, 194*
D2(185) = 185, 188*
D2(192) = 192, 199, 202, 206*
D2(195) = 195, 199*
D2(197) = 197, 200, 206*
D2(204) = 204, 211, 214, 218*
D2(207) = 207, 211*
D2(209) = 209, 212, 217*
D2(215) = 215, 218*
D2(216) = 216, 223, 226*
D2(219) = 219, 223*
D2(221) = 221, 224, 229, 232,
237, 241, 244, 249, 253, 256, 261, 265, 268, 273, 277*
D2(225) = 225, 229*
D2(228) = 228, 235, 238, 242*
D2(231) = 231, 235*
D2(233) = 233, 236, 241*
D2(240) = 240, 249*
D2(243) = 243, 247, 250, 254*
D2(245) = 245, 248, 253*
D2(252) = 252, 260, 266*
D2(255) = 255, 259, 262, 266*
D2(257) = 257, 260*
D2(264) = 264, 271, 274, 278,
282, 287, 290*
D2(267) = 267, 271*
D2(269) = 269, 272, 277*
D2(275) = 275, 278*
D2(276) = 276, 283, 286*
D2(279) = 279, 283*
D2(281) = 281, 284, 289, 292,
297, 301, 304, 309, 313, 316, 321, 325, 328, 333, 337*
D2(285) = 285, 289*
D2(288) = 288, 295, 298, 302*
D2(291) = 291, 295*
D2(293) = 293, 296, 301*
D2(300) = 300, 308, 313*
D2(303) = 303, 307, 310, 314*
D2(305) = 305, 308*
D2(312) = 312, 319, 322, 326*
D2(315) = 315, 319*
D2(317) = 317, 320, 326*
D2(324) = 324, 331, 334, 338,
342, 347, 350*
D2(327) = 327, 331*
D2(329) = 329, 332, 337*
D2(335) = 335, 338*
D2(336) = 336, 345, 349, 352,
357, 361, 364, 369, 373, 376, 381, 385, 388, 393, 397*
D2(339) = 339, 343, 346*
D2(341) = 341, 344, 349*
D2(348) = 348, 355, 358, 362*
D2(351) = 351, 355*
D2(353) = 353, 356, 361*
D2(360) = 360, 371*
D2(363) = 363, 367, 370, 374*
D2(365) = 365, 368, 373*
D2(372) = 372, 379, 382, 386*
D2(375) = 375, 379*
D2(377) = 377, 380, 386*
D2(384) = 384, 391, 394, 398,
402, 407, 410*
D2(387) = 387, 391*
D2(389) = 389, 392, 397*
D2(395) = 395, 398*
D2(396) = 396, 403, 406*
D2(399) = 399, 403*
D2(401) = 401, 404, 409, 412,
417, 421, 424, 429, 433, 436, 441, 445, 448, 453, 457*
D2(405) = 405, 409*
D2(408) = 408, 415, 418, 422*
D2(411) = 411, 415*
D2(413) = 413, 416, 421*
D2(420) = 420, 429*
D2(423) = 423, 427, 430, 434*
D2(425) = 425, 428, 433*
D2(432) = 432, 439, 442, 446*
D2(435) = 435, 439*
D2(437) = 437, 440, 446*
D2(444) = 444, 451, 454, 458,
462, 467, 470*
D2(447) = 447, 451*
D2(449) = 449, 452, 457*
D2(455) = 455, 458*
D2(456) = 456, 463, 466*
D2(459) = 459, 463*
D2(461) = 461, 464, 469, 472,
477, 481, 484, 489, 493, 496, 501, 505, 508, 513, 517*
D2(465) = 465, 469*
D2(468) = 468, 475, 478, 482*
D2(471) = 471, 475*
D2(473) = 473, 476, 481*
D2(480) = 480, 489*
D2(483) = 483, 487, 490, 494*
D2(485) = 485, 488, 493*
D2(492) = 492, 499, 502, 506*
D2(495) = 495, 499*
D2(497) = 497, 500, 506*
D2(504) = 504, 514, 518, 522,
527, 530*
D2(507) = 507, 511, 514*
D2(509) = 509, 512, 517*
D2(515) = 515, 518*
D2(516) = 516, 523, 526*
D2(519) = 519, 523*
D2(521) = 521, 524, 529, 532,
537, 541, 544, 549, 553, 556, 561, 565, 568, 573, 577*
D2(525) = 525, 529*
D2(528) = 528, 535, 538, 542*
D2(531) = 531, 535*
D2(533) = 533, 536, 541*
D2(540) = 540, 548, 553*
D2(543) = 543, 547, 550, 554*
D2(545) = 545, 548*
D2(552) = 552, 559, 562, 566*
D2(555) = 555, 559*
D2(557) = 557, 560, 566*
D2(564) = 564, 571, 574, 578,
582, 587, 590*
D2(567) = 567, 571*
D2(569) = 569, 572, 577*
D2(575) = 575, 578*
D2(576) = 576, 583, 586*
D2(579) = 579, 583*
D2(581) = 581, 584, 589, 592,
597, 601, 604, 609, 613, 616, 621, 625, 628, 633, 637, 640, 646, 650*
D2(585) = 585, 589*
D2(588) = 588, 596, 601*
D2(591) = 591, 595, 598, 602*
D2(593) = 593, 596*
D2(600) = 600, 609*
D2(603) = 603, 607, 610, 614*
D2(605) = 605, 608, 613*
D2(612) = 612, 619, 622, 626*
D2(615) = 615, 619*
D2(617) = 617, 620, 626*
D2(624) = 624, 631, 634, 638*
D2(627) = 627, 631*
D2(629) = 629, 632, 637*
D2(635) = 635, 638*
D2(636) = 636, 643, 646*
D2(639) = 639, 643*
D2(641) = 641, 644, 649, 652,
657, 661, 664, 669, 673, 676, 681, 685, 688, 693, 697*
D2(645) = 645, 649*
D2(648) = 648, 655, 658, 662*
D2(651) = 651, 655*
D2(653) = 653, 656, 661*
D2(660) = 660, 668, 673*
D2(663) = 663, 667, 670, 674*
D2(665) = 665, 668*
D2(672) = 672, 681*
D2(675) = 675, 679, 682, 686*
D2(677) = 677, 680, 686*
D2(684) = 684, 691, 694, 698,
702, 707, 710*
D2(687) = 687, 691*
D2(689) = 689, 692, 697*
D2(695) = 695, 698*
D2(696) = 696, 703, 706*
D2(699) = 699, 703*
D2(701) = 701, 704, 709, 712,
717, 721, 724, 729, 733, 736, 741, 745, 748, 753, 757*
D2(705) = 705, 709*
D2(708) = 708, 715, 718, 722*
D2(711) = 711, 715*
D2(713) = 713, 716, 721*
D2(720) = 720, 731*
D2(723) = 723, 727, 730, 734*
D2(725) = 725, 728, 733*
D2(732) = 732, 739, 742, 746*
D2(735) = 735, 739*
D2(737) = 737, 740, 746*
D2(744) = 744, 751, 754, 758,
762, 767, 770*
D2(747) = 747, 751*
D2(749) = 749, 752, 757*
D2(755) = 755, 758*
D2(756) = 756, 764, 769, 772,
777, 781, 784, 789, 793, 796, 801, 805, 808, 813, 817*
D2(759) = 759, 763, 766*
D2(761) = 761, 764*
D2(765) = 765, 769*
D2(768) = 768, 775, 778, 782*
D2(771) = 771, 775*
D2(773) = 773, 776, 781*
D2(780) = 780, 788, 793*
D2(783) = 783, 787, 790, 794*
D2(785) = 785, 788*
D2(792) = 792, 799, 802, 806*
D2(795) = 795, 799*
D2(797) = 797, 800, 806*
D2(804) = 804, 811, 814, 818,
822, 827, 830*
D2(807) = 807, 811*
D2(809) = 809, 812, 817*
D2(815) = 815, 818*
D2(816) = 816, 823, 826*
D2(819) = 819, 823*
D2(821) = 821, 824, 829, 832,
837, 841, 844, 849, 853, 856, 861, 865, 868, 873, 877*
D2(825) = 825, 829*
D2(828) = 828, 835, 838, 842*
D2(831) = 831, 835*
D2(833) = 833, 836, 841*
D2(840) = 840, 851*
D2(843) = 843, 847, 850, 854*
D2(845) = 845, 848, 853*
D2(852) = 852, 859, 862, 866*
D2(855) = 855, 859*
D2(857) = 857, 860, 866*
D2(864) = 864, 871, 874, 878,
882, 887, 890*
D2(867) = 867, 871*
D2(869) = 869, 872, 877*
D2(875) = 875, 878*
D2(876) = 876, 883, 886*
D2(879) = 879, 883*
D2(881) = 881, 884, 889, 892,
897, 901, 904, 909, 913, 916, 921, 925, 928, 933, 937*
D2(885) = 885, 889*
D2(888) = 888, 895, 898, 902*
D2(891) = 891, 895*
D2(893) = 893, 896, 901*
D2(900) = 900, 908, 913*
D2(903) = 903, 907, 910, 914*
D2(905) = 905, 908*
D2(912) = 912, 919, 922, 926*
D2(915) = 915, 919*
D2(917) = 917, 920, 926*
D2(924) = 924, 932, 937*
D2(927) = 927, 931, 934, 938,
942, 947, 950*
D2(929) = 929, 932*
D2(935) = 935, 938*
D2(936) = 936, 943, 946*
D2(939) = 939, 943*
D2(941) = 941, 944, 949, 952,
957, 961, 964, 969, 973, 976, 981, 985, 988, 993, 997*
D2(945) = 945, 949*
D2(948) = 948, 955, 958, 962*
D2(951) = 951, 955*
D2(953) = 953, 956, 961*
D2(960) = 960, 969*
D2(963) = 963, 967, 970, 974*
D2(965) = 965, 968, 973*
D2(972) = 972, 979, 982, 986*
D2(975) = 975, 979*
D2(977) = 977, 980, 986*
D2(984) = 984, 991, 994, 998,
1002, 1007, 1010*
D2(987) = 987, 991*
D2(989) = 989, 992, 997*
D2(995) = 995, 998*
D2(996) = 996, 1003, 1006*
D2(999) = 999, 1003*
D2(1001) = 1001, 1004, 1009, 1012, 1017, 1021, 1024, 1029, 1033,
1036, 1041, 1045, 1048, 1053, 1057, 1060, 1066, 1070*
...
D2(1961) = 1961, 1964, 1969, 1972, 1977, 1981, 1984, 1989, 1993,
1996, 2001, 2005, 2008, ...
On pourrait dessiner un début de carte des suites D2(n)
(la couleur jaune portée sur certains nombres est expliquée plus bas) :
1--4--9---13--16--21--25--28--33--37--40
...
| |
5--8---+ 29--32--+ 41 ...
|
2--6--11--14--18--23--26--30------+
| |
3--7--10--+ 17--20--+ 36--43 ...
| |
12--19--22--+ 39--+
|
15--+ 24--31--34--38--42 ...
| |
27--+ 35--+
Plusieurs questions se posent :
- Peut-on calculer le millième terme d’une suite D2 sans en calculer tous les
autres ?
- On se donne un nombre (2008, par exemple) : peut-on trouver
le premier terme de la plus longue suite aboutissant à ce nombre ? [Jacques T. a calculé que 2008 ne faisait partie que
d’une seule suite D2 — celle qui
commence par 1961 : D2(1961) = 1961, 1964, 1969, 1972, 1977, 1981, 1984, 1989, 1993,
1996, 2001, 2005, 2008,...]
- Certains nombres n’ont pas de
prédécesseur, ce sont ceux qui ouvrent une nouvelle suite D2 (ils sont surlignés de jaune sur la carte ci-dessus) ; à quelle loi obéissent ces nombres ?
- Qu’en est-il des nombres qui ont un seul
prédécesseur, de ceux qui en ont exactement deux, de ceux qui en ont exactement
trois, etc. ? (13 a deux prédécesseurs sur la carte des suites D2, ce sont 8 et 9 ; 37 a exactement trois
prédécesseurs : 30, 32 et 33)
Jacques T. a calculé la
série des nombres D2 qui n’ont pas
de prédécesseur (cette suite est composée du premier terme de chaque suite du
long tableau vertical « avec astérisque », plus haut) :
AucunPrédécesseur(D2) = 1, 2, 3, 5, 12, 15, 17, 24, 27, 29, 35, 36, 39, 41,
45, 48, 51, 53, 60, 63, 65, 72, 75, 77, 84, 87, 89, 95, 96, 99, 101, 105, 108,
111, 113, 120, 123, 125, 132, 135, 137, 144, 147, 149, 155, 156, 159, 161, 165,
168, 171, 173, 180, 183, 185, 192, 195, 197, 204, 207, 209, 215, 216, 219,
221, 225, 228, 231, 233, 240, 243, 245, 252, 255, 257, 264, 267, 269, 275, 276,
279, 281, 285, 288, 291, 293, 300, 303, 305, 312, 315, 317, 324, 327, 329, 335,
336, 339, 341, 348, 351, 353, 360, 363, 365, 372, 375, 377, 384, 387, 389, 395,
396, 399, 401, 405, 408, 411, 413, 420, 423, 425, 432, 435, 437, 444, 447, 449,
455, 456, 459, 461, 465, 468, 471, 473, 480, 483, 485, 492, 495, 497, 504, 507,
509, 515, 516, 519, 521, 525, 528, 531, 533, 540, 543, 545, 552, 555, 557, 564,
567, 569, 575, 576, 579, 581, 585, 588, 591, 593, 600, 603, 605, 612, 615, 617,
624, 627, 629, 635, 636, 639, 641, 645, 648, 651, 653, 660, 663, 665, 672, 675,
677, 684, 687, 689, 695, 696, 699, 701, 705, 708, 711, 713, 720, 723, 725, 732,
735, 737, 744, 747, 749, 755, 756, 759, 761, 765, 768, 771, 773, 780, 783, 785,
792, 795, 797, 804, 807, 809, 815, 816, 819, 821, 825, 828, 831, 833, 840, 843,
845, 852, 855, 857, 864, 867, 869, 875, 876, 879, 881, 885, 888, 891, 893, 900,
903, 905, 912, 915, 917, 924, 927, 929, 935, 936, 939, 941, 945, 948, 951, 953,
960, 963, 965, 972, 975, 977, 984, 987, 989, 995, 996, 999, 1001, ...
Voici le début de la suite commençant par
1 et obéissant à la règle D3
(« Ajouter à n le 3e
nombre qui ne divise pas n »),
merci à Nicolas Graner pour ses
calculs :
D3(1) = 1, 5, 9, 14, 19, 23, 27, 32, 38, 43,
47, 51, 56, 62, 67, 71, 75, 81, 86, 91, 95, 99, 104, 110, 116, 122, 127, 131,
135, 141, 146, 151, 155, 159, 164, 170, 176, 182, 187, 191, 195, 201, 206, 211,
215, 219, 224, 230, 236, 242, 247, 251, 255, 261, 266, 271, 275, 279, 284, 290,
296, 302, 307, 311, 315, 321, 326, 331, 335, 339, 344, 350, 356, 362, 367, 371,
375, 381, 386, 391, 395, 399, 404, 410, 416, 422, 427, 431, 435, 441, 446, 451,
455, 459, 464, 470, 476, 482, 487, 491, 495, 501, 506, 511, 515, 519, 524, 530,
536, 542, 547, 551, 555, 561, 566, 571, 575, 579, 584, 590, 596, 602, 607, 611,
615, 621, 626, 631, 635, 639, 644, 650, 656, 662, 667, 671, 675, 681, 686, 691,
695, 699, 704, 710, 716, 722, 727, 731, 735, 741, 746, 751, 755, 759, 764, 770,
776, 782, 787, 791, 795, 801, 806, 811, 815, 819, 824, 830, 836, 842, 847, 851,
855, 861, 866, 871, 875, 879, 884, 890, 896, 902, 907, 911, 915, 921, 926, 931,
935, 939, 944, 950, 956, 962, 967, 971, 975, 981, 986, 991, 995, 999, 1004,
1010, 1016, 1022, 1027, 1031, 1035, 1041, 1046, 1051, 1055, 1059, 1064, 1070,
1076, 1082, 1087, 1091, 1095, 1101, 1106, 1111, 1115, 1119, 1124, 1130, 1136,
1142, 1147, 1151, 1155, 1161, 1166, 1171, 1175, 1179, 1184, 1190, 1196, 1202,
1207, 1211, 1215, 1221, 1226, 1231, 1235, 1239, 1244, 1250, 1256, 1262, 1267,
1271, 1275, 1281, 1286, 1291, 1295, 1299, 1304, 1310, 1316, 1322, 1327, 1331,
1335, 1341, 1346, 1351, 1355, 1359, 1364, 1370, 1376, 1382, 1387, 1391, 1395,
1401, 1406, 1411, 1415, 1419, 1424, 1430, 1436, 1442, 1447, 1451, 1455, 1461,
1466, 1471, 1475, 1479, 1484, 1490, 1496, 1502, 1507, 1511, 1515, 1521, 1526,
1531, 1535, 1539, 1544, 1550, 1556, 1562, 1567, 1571, 1575, 1581, 1586, 1591,
1595, 1599, 1604, 1610, 1616, 1622, 1627, 1631, 1635, 1641, 1646, 1651, 1655,
1659, 1664, 1670, 1676, 1682, 1687, 1691, 1695, 1701, 1706, 1711, 1715, 1719,
1724, 1730, 1736, 1742, 1747, 1751, 1755, 1761, 1766, 1771, 1775, 1779, 1784,
1790, 1796, 1802, 1807, 1811, 1815, 1821, 1826, 1831, 1835, 1839, 1844, 1850,
1856, 1862, 1867, 1871, 1875, 1881, 1886, 1891, 1895, 1899, 1904, 1910, 1916, 1922,
1927, 1931, 1935, 1941, 1946, 1951, 1955, 1959, 1964, 1970, 1976, 1982, 1987,
1991, 1995, 2001, 2006, 2011, 2015, 2019, 2024, 2030, 2036, 2042, 2047, 2051,
2055, 2061, 2066, 2071, 2075, 2079, 2084, 2090, 2096, 2102, 2107, 2111, 2115,
2121, 2126, 2131, 2135, 2139, 2144, 2150, 2156, 2162, 2167, 2171, 2175, 2181,
2186, 2191, 2195, 2199, 2204, 2210, 2216, 2222, 2227, 2231, 2235, 2241, 2246,
2251, 2255, 2259, 2264, 2270, 2276, 2282, 2287, 2291, 2295, 2301, 2306, 2311,
2315, 2319, 2324, 2330, 2336, 2342, 2347, 2351, 2355, 2361, 2366, 2371, 2375,
2379, 2384, 2390, 2396, 2402, 2407, 2411, 2415, 2421, 2426, 2431, 2435, 2439,
2444, 2450, 2456, 2462, 2467, 2471, 2475, 2481, 2486, 2491, 2495, 2499, 2504,
2510, 2516, 2522, 2527, 2531, 2535, 2541, 2546, 2551, 2555, 2559, 2564, 2570,
2576, 2582, 2587, 2591, 2595, 2601, 2606, 2611, 2615, 2619, 2624, 2630, 2636,
2642, 2647, 2651, 2655, 2661, 2666, 2671, 2675, 2679, 2684, 2690, 2696, 2702,
2707, 2711, 2715, 2721, 2726, 2731, 2735, 2739, 2744, 2750, 2756, 2762, 2767,
2771, 2775, 2781, 2786, 2791, 2795, 2799, 2804, 2810, 2816, 2822, 2827, 2831,
2835, 2841, 2846, 2851, 2855, 2859, 2864, 2870, 2876, 2882, 2887, 2891, 2895,
2901, 2906, 2911, 2915, 2919, 2924, 2930, 2936, 2942, 2947, 2951, 2955, 2961,
2966, 2971, 2975, 2979, 2984, 2990, 2996, 3002, 3007, 3011, 3015, 3021, 3026,
3031, 3035, 3039, 3044, 3050, 3056, 3062, 3067, 3071, 3075, 3081, 3086, 3091,
3095, 3099, 3104, 3110, 3116, 3122, 3127, 3131, 3135, 3141, 3146, 3151, 3155,
3159, 3164, 3170, 3176, 3182, 3187, 3191, 3195, 3201, 3206, 3211, 3215, 3219,
3224, 3230, 3236, 3242, 3247, 3251, 3255, 3261, 3266, 3271, 3275, 3279, 3284,
3290, 3296, 3302, 3307, 3311, 3315, 3321, 3326, 3331, 3335, 3339, 3344, 3350,
3356, 3362, 3367, 3371, 3375, 3381, 3386, 3391, 3395, 3399, 3404, 3410, 3416,
3422, 3427, 3431, 3435, 3441, 3446, 3451, 3455, 3459, 3464, 3470, 3476, 3482,
3487, 3491, 3495, 3501, 3506, 3511, 3515, 3519, 3524, 3530, 3536, 3542, 3547,
3551, 3555, 3561, 3566, 3571, 3575, 3579, 3584, 3590, 3596, 3602, 3607, 3611,
3615, 3621, 3626, 3631, 3635, 3639, 3644, 3650, 3656, 3662, 3667, 3671, 3675,
3681, 3686, 3691, 3695, 3699, 3704, 3710, 3716, 3722, 3727, 3731, 3735, 3741,
3746, 3751, 3755, 3759, 3764, 3770, 3776, 3782, 3787, 3791, 3795, 3801, 3806,
3811, 3815, 3819, 3824, 3830, 3836, 3842, 3847, 3851, 3855, 3861, 3866, 3871,
3875, 3879, 3884, 3890, 3896, 3902, 3907, 3911, 3915, 3921, 3926, 3931, 3935,
3939, 3944, 3950, 3956, 3962, 3967, 3971, 3975, 3981, 3986, 3991, 3995, 3999,
4004, 4010, 4016, 4022, 4027, 4031, 4035, 4041, 4046, 4051, 4055, 4059, 4064,
4070, 4076, 4082, 4087, 4091, 4095, 4101, 4106, 4111, 4115, 4119, 4124, 4130,
4136, 4142, 4147, 4151, 4155, 4161, 4166, 4171, 4175, 4179, 4184, 4190, 4196,
4202, 4207, 4211, 4215, 4221, 4226, 4231, 4235, 4239, 4244, 4250, 4256, 4262,
4267, 4271, 4275, 4281, 4286, 4291, 4295, 4299, 4304, 4310, 4316, 4322, 4327,
4331, 4335, 4341, 4346, 4351, 4355, 4359, 4364, 4370, 4376, 4382, 4387, 4391,
4395, 4401, 4406, 4411, 4415, 4419, 4424, 4430, 4436, 4442, 4447, 4451, 4455,
4461, 4466, 4471, 4475, 4479, 4484, 4490, 4496, 4502, 4507, 4511, 4515, 4521,
4526, 4531, 4535, 4539, 4544, 4550, 4556, 4562, 4567, 4571, 4575, 4581, 4586,
4591, 4595, 4599, 4604, 4610, 4616, 4622, 4627, 4631, 4635, 4641, 4646, 4651,
4655, 4659, 4664, 4670, 4676, 4682, 4687, 4691, 4695, 4701, 4706, 4711, 4715,
4719, 4724, 4730, 4736, 4742, 4747, 4751, 4755, 4761, 4766, 4771, 4775, 4779,
4784, 4790, 4796, 4802, 4807, 4811, 4815, 4821, 4826, 4831, 4835, 4839, 4844,
4850, 4856, 4862, 4867, 4871, 4875, 4881, 4886, 4891, 4895, 4899, 4904, 4910,
4916, 4922, 4927, 4931, 4935, 4941, 4946, 4951, 4955, 4959, 4964, 4970, 4976,
4982, 4987, 4991, ...
Un début de cartographie des suites D3(n) :
1--5--9------14--19--23--27--32------38--43--47--51--56------62--67--71------75--81--86--91--95--99
...
| | | |
| |
3--8------+ |
30------+ |
| |
| | |
| |
12------20------+ | |
| |
| | |
| |
4-----10--16------22--+ | |
| |
| |
| |
18--25--29--33------+ |
| |
| | | |
| 40------+
| |
| | |
| 46--+ |
| | |
24------+
37--41--45------+ |
|
36------+ |
|
|
2--7--11--15--21------26--31--35--39--44------50--+
| | |
6--13--17--+ 28------34--+
42------+
La suite des nombres qui n’ont pas de prédécesseur dans D3 (en jaune ci-dessus) est :
AucunPrédécesseur(D3)
= 1, 2, 3, 4, 6, 12, 18, 24, 28, 30, 36, 37, 40, 42, 46, 48, 49, 52, 54, 55,
60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 97, 100, 102, 106, 108, 112, 114, 115,
120, 124, 126, 132, 133, 138, 144, 150, 156, 157, 160, 162, 166, 168, 172, 174,
175, 180, 184, 186, 192, 198, 200, 204, 208, 210, 216, 217, 220, 222, 228, 232,
234, 235, 240, 244, 246, 252, 258, 260, 264, 268, 270, 276, 277, 280, 282, 286,
288, 292, 294, 295, 300, 301, 304, 306, 312, 318, 320, 324, 328, 330, 336, 337,
340, 342, 348, 352, 354, 355, 360, 364, 366, 372, 378, 384, 385, 388, 390, 396,
397, 400, 402, 406, 408, 412, 414, 415, 420, 424, 426, 432, 438, 440, 444, 448,
450, 456, 457, 460, 462, 466, 468, 469, 472, 474, 475, 480, 484, 486, 492, 498,
504, 508, 510, 516, 517, 520, 522, 526, 528, 532, 534, 535, 540, 544, 546, 552,
553, 558, 560, 564, 568, 570, 576, 577, 580, 582, 588, 592, 594, 595, 600, 604,
606, 612, 618, 624, 628, 630, 636, 637, 640, 642, 646, 648, 652, 654, 655, 660,
664, 666, 672, 678, 680, 684, 688,
690, 696, 697, 700, 702, 706, 712, 714, 715, 720, 721, 724, 726, 732, 738, 744,
748, 750, 756, 757, 760, 762, 768, 772, 774, 775, 780, 784, 786, 792, 798, 800,
804, 805, 808, 810, 816, 817, 820, 822, 826, 828, 832, 834, 835, 840, 844, 846,
852, 858, 864, 868, 870, 876, 877, 880, 882, 886, 888, 889, 892, 894, 895, 900,
904, 906, 912, 918, 920, 924, 928, 930, 936, 937, 940, 942, 948, 952, 954, 955,
960, 964, 966, 972, 973, 978, 984, 990, 996, 997, 1000, 1002, 1006, 1008, 1012,
1014, 1015, 1020, 1024, 1026, 1032, 1038, 1040, 1044, 1048, 1050, 1056, 1057,
1060, 1062, 1066, 1068, 1072, 1074, 1075, 1080, 1084, 1086, 1092, 1098, 1100,
1104, 1108, 1110, 1116, 1117, 1120, 1122, 1126, 1128, 1132, 1134, 1135, 1140,
1141, 1144, 1146, 1152, 1158, 1160, 1164, 1168, 1170, 1176, 1177, 1180, 1182,
1188, 1192, 1194, 1195, 1200, 1204, 1206, 1212, 1218, 1224, 1225, 1228, 1230,
1236, 1237, 1240, 1242, 1246, 1248, 1252, 1254, 1255, 1260, 1264, 1266, 1272,
1278, 1280, 1284, 1288, 1290, 1296, 1297, 1300, 1302, 1308, 1309, 1312, 1314,
1315, 1320, 1324, 1326, 1332, 1338, 1344, 1348, 1350, 1356, 1357, 1360, 1362,
1366, 1368, 1372, 1374, 1375, 1380, 1384, 1386, 1392, 1393, 1398, 1400, 1404,
1408, 1410, 1416, 1417, 1420, 1422, 1426, 1428, 1432, 1434, 1435, 1440, 1444,
1446, 1452, 1458, 1464, 1468, 1470, 1476, 1477, 1480, 1482, 1486, 1488, 1492,
1494, 1495, 1500, 1504, 1506, 1512, 1518, 1520, 1524, 1528, 1530, 1536, 1537,
1540, 1542, 1546, 1552, 1554, 1555, 1560, 1561, 1564, 1566, 1572, 1578, 1584,
1588, 1590, 1596, 1597, 1600, 1602, 1606, 1608, 1612, 1614, 1615, 1620, 1624,
1626, 1632, 1638, 1640, 1644, 1645, 1648, 1650, 1656, 1657, 1660, 1662, 1668,
1672, 1674, 1675, 1680, 1684, 1686, 1692, 1698, 1704, 1708, 1710, 1716, 1717,
1720, 1722, 1726, 1728, 1729, 1732, 1734, 1735, 1740, 1744, 1746, 1752, 1758,
1760, 1764, 1768, 1770, 1776, 1777, 1780, 1782, 1786, 1788, 1792, 1794, 1795,
1800, 1804, 1806, 1812, 1818, 1824, 1830, 1836, 1837, 1840, 1842, 1846, 1848,
1852, 1854, 1855, 1860, 1864, 1866, 1872, 1878, 1880, 1884, 1888, 1890, 1896, 1897,
1900, 1902, 1906, 1908, 1912, 1914, 1915, 1920, 1924, 1926, 1932, 1938, 1940,
1944, 1948, 1950, 1956, 1957, 1960, 1962, 1966, 1968, 1972, 1974, 1975, 1980,
1981, 1984, 1986, 1992, 1998, 2000, 2004, 2008, ...
On voit que 2008 fait partie de cette liste ;
il est donc impossible d’atteindre ce nombre selon la loi D3 (alors que c’est possible avec D2).
Jacques T. remarque aussi que les
nombres de AucunPrédécesseur(D3)
sont presque tous divisibles par 6 : « The above
numbers are 'nearly' all
multiple of 6 ».
__________
Note vaguement autoréférente :
La loi D2 qui ouvre
cette page « Ajouter à n le
2e nombre qui ne divise pas n »
pourrait se parfumer d’autoréférence : « Ajouter à n le ne nombre qui ne divise
pas n ». Ainsi, au lieu
d’ajouter à n le 2e entier
qui ne le divise pas, ou le 3e, ou le 4e, etc. (ce sont
les suites D2, D3, D4...), on lui ajouterait chaque fois le ne
entier qui ne le divise pas ; c’est donc n lui-même qui fixerait le nombre de non-diviseurs à considérer.
Appelons DN(1) la suite de tels
nombres qui commence par 1 ; elle a été calculée indépendamment par Jacques T., Jean-Marc Falcoz Nathalie Demole et Hans Havermann (merci à eux) :
DN(1) = 1, 3, 8, 20, 46, 96, 204,
420, 864, 1752, 3520, 7068, 14160, 28360, 56736, 113508, 227040, 454176,
908424, 1816944, 3633908, 7267828, 14535662, 29071328, 58142704, 116285418,
232570884, 465141864, 930283760, 1860567600, 3721135320, 7442270736, 14884541492,
29769083008, 59538166080, 119076332272, 238152664554, 476305329172,
952610658350, 1905221316724, 3810442633460, 7620885266932, 15241770533876,
30483541067764, 60967082135576, 121934164271216, 243868328542452,
487736657084916, 975473314169850, 1950946628339784, 3901893256679600,
7803786513359440, 15607573026718920, 31215146053437936, 62430292106875932,
124860584213751888, 249721168427503836, 499442336855007768, 998884673710015632,
1997769347420031424, 3995538694840062876, 7991077389680125800, 15982154779360251888,
31964309558720504176, 63928619117441008362, 127857238234882016772,
255714476469764033736, 511428952939528067568, 1022857905879056135176,
2045715811758112270368, 4091431623516224540760, 8182863247032449081584,
16365726494064898163188, 32731452988129796326472, 65462905976259592652952,
130925811952519185306032, 261851623905038370612104, 523703247810076741224216,
1047406495620153482448496, 2094812991240306964897012,
4189625982480613929794036, 8379251964961227859588084,
16758503929922455719176192, 33517007859844911438352480,
67034015719689822876705056, 134068031439379645753410160,
268136062878759291506820640, 536272125757518583013641472,
1072544251515037166027283088, 2145088503030074332054566196,
4290177006060148664109132404, 8580354012120297328218264820,
17160708024240594656436529688, 34321416048481189312873059408,
68642832096962378625746118936, 137285664193924757251492238000,
274571328387849514502984477280, 549142656775699029005968954848,
1098285313551398058011937909732, 2196570627102796116023875819704, 4393141254205592232047751639472,
8786282508411184464095503278954, 17572565016822368928191006558100,
35145130033644737856382013117352, 70290260067289475712764026235216,
140580520134578951425528052470512, 281161040269157902851056104941104,
562322080538315805702112209882448, 1124644161076631611404224419764936,
2249288322153263222808448839530000, 4498576644306526445616897679060100,
8997153288613052891233795358120488, 17994306577226105782467590716241104,
35988613154452211564935181432482288, 71977226308904423129870362864964616,
143954452617808846259740725729929616, 287908905235617692519481451459859312,
575817810471235385038962902919718684, 1151635620942470770077925805839437380,
2303271241884941540155851611678874808, 4606542483769883080311703223357749680,
9213084967539766160623406446715499840, 18426169935079532321246812893430999736,
36852339870159064642493625786861999480, 73704679740318129284987251573723999088,
147409359480636258569974503147447998336,
294818718961272517139949006294895996704, 589637437922545034279898012589791993792,
1179274875845090068559796025179583987640,
2358549751690180137119592050359167975792,
4717099503380360274239184100718335951904,
9434199006760720548478368201436671904000,
18868398013521441096956736402873343808576, 37736796027042882193913472805746687617208,
75473592054085764387826945611493375234928,
150947184108171528775653891222986750469876,
301894368216343057551307782445973500939944,
603788736432686115102615564891947001879952,
1207577472865372230205231129783894003759944, 2415154945730744460410462259567788007520016,
4830309891461488920820924519135576015040672,
9660619782922977841641849038271152030081392,
19321239565845955683283698076542304060163104,
38642479131691911366567396153084608120326592,
77284958263383822733134792306169216240653408, 154569916526767645466269584612338432481307008,
309139833053535290932539169224676864962614400,
618279666107070581865078338449353729925231872,
1236559332214141163730156676898707459850464032,
2473118664428282327460313353797414919700928256, 4946237328856564654920626707594829839401856944,
9892474657713129309841253415189659678803715808,
19784949315426258619682506830379319357607432384,
39569898630852517239365013660758638715214865440,
79139797261705034478730027321517277430429730976,
158279594523410068957460054643034554860859463104,
316559189046820137914920109286069109721718926432,
633118378093640275829840218572138219443437852876,
1266236756187280551659680437144276438886875705800,
2532473512374561103319360874288552877773751413136,
5064947024749122206638721748577105755547502826312,
10129894049498244413277443497154211511095005652672,
20259788098996488826554886994308423022190011305568,
40519576197992977653109773988616846044380022611160,
81039152395985955306219547977233692088760045222336,
162078304791971910612439095954467384177520090444784,
324156609583943821224878191908934768355040180890208,
648313219167887642449756383817869536710080361780608,
1296626438335775284899512767635739073420160723561984,
2593252876671550569799025535271478146840321447124288, 5186505753343101139598051070542956293680642894248632,
10373011506686202279196102141085912587361285788497328,
20746023013372404558392204282171825174722571576994816,
41492046026744809116784408564343650349445143153989676,
82984092053489618233568817128687300698890286307979448, 165968184106979236467137634257374601397780572615958912,
331936368213958472934275268514749202795561145231917984,
663872736427916945868550537029498405591122290463836064,
1327745472855833891737101074058996811182244580927672224,
2655490945711667783474202148117993622364489161855344544,
5310981891423335566948404296235987244728978323710689472,
10621963782846671133896808592471974489457956647421379056,
21243927565693342267793617184943948978915913294842758192,
42487855131386684535587234369887897957831826589685516464, 84975710262773369071174468739775795915663653179371033728,
169951420525546738142348937479551591831327306358742067584,
339902841051093476284697874959103183662654612717484135424,
679805682102186952569395749918206367325309225434968271200,
1359611364204373905138791499836412734650618450869936565440,
2719222728408747810277582999672825469301236901739873131216,
5438445456817495620555165999345650938602473803479746262752,
10876890913634991241110331998691301877204947606959492525888,
21753781827269982482220663997382603754409895213918985051832,
43507563654539964964441327994765207508819790427837970103792,
87015127309079929928882655989530415017639580855675940207664,
174030254618159859857765311979060830035279161711351880415368,
348060509236319719715530623958121660070558323422703760830992,
696121018472639439431061247916243320141116646845407521662064,
1392242036945278878862122495832486640282233293690815043324148,
2784484073890557757724244991664973280564466587381630086648344,
5568968147781115515448489983329946561128933174763260173296720,
11137936295562231030896979966659893122257866349526520346593760,
22275872591124462061793959933319786244515732699053040693187568,
44551745182248924123587919866639572489031465398106081386375776,
89103490364497848247175839733279144978062930796212162772752320,
178206980728995696494351679466558289956125861592424325545507328,
356413961457991392988703358933116579912251723184848651091014816,
712827922915982785977406717866233159824503446369697302182029824,
1425655845831965571954813435732466319649006892739394604364059728,
2851311691663931143909626871464932639298013785478789208728119536,
5702623383327862287819253742929865278596027570957578417456239232,
11405246766655724575638507485859730557192055141915156834912478496,
22810493533311449151277014971719461114384110283830313669824957088,
45620987066622898302554029943438922228768220567660627339649914320,
91241974133245796605108059886877844457536441135321254679299832480,...
La formule donnant le terme a(n) ci-dessus est très simple, comme
l’ont noté plusieurs intervenants sur SeqFans : « Each term is twice the previous term plus the number of divisors of the previous term » ou, de manière plus concise : « This rule simply takes
n to 2n + tau(n) ».
Franklin T. Adams-Watters ajoute : « (...) it
is likely that 1 is the only
square in the sequence, which
would make 1 and 3 the only odd numbers
in the sequence. » Cette suite vaguement autoréférente
est désormais archivée dans la base de Neil
Sloane sous l’étiquette A140781.
Les nombres DN qui n’ont pas de prédécesseurs sont (merci encore à
Nicolas Graner) :
AucunPrédécesseur(DN) = 1, 2, 4, 5, 7, 9, 10, 13, 14, 15, 17, 18, 19, 22, 23,
25, 26, 27, 29, 31, 33, 35, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 52, 54, 55,
57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90,
91, 93, 95, 97, 98, 99, 100, 102, 103, 104, 105, 107, 109, 111, 112, 113, 115,
117, 119, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 137, 138, 139, 141,
143, 145, 146, 147, 149, 150, 151, 153, 154, 155, 157, 159, 161, 162, 163, 165,
166, 169, 171, 172, 173, 175, 177, 179, 181, 182, 183, 185, 187, 188, 189, 191,
193, 195, 197, 198, 199, 200, 201, 203, 205, 206, 207, 210, 211, 213, 214, 215,
217, 219, 221, 222, 223, 224, 225, 227, 229, 230, 231, 232, 233, 235, 237, 239,
241, 243, 244, 246, 247, 249, 251, 252, 253, 255, 257, 258, 259, 260, 261, 263,
265, 266, 267, 269, 271, 273, 274, 275, 277, 279, 281, 282, 283, 285, 287, 289,
291, 293, 295, 297, 298, 299, 301, 305, 306, 307, 308, 309, 310, 311, 313, 315,
317, 318, 319, 321, 323, 325, 327, 329, 330, 331, 333, 335, 337, 339, 340, 342,
343, 344, 345, 346, 347, 349, 351, 353, 354, 355, 357, 359, 361, 363, 365, 366,
367, 368, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387, 389, 390, 391, 393,
394, 395, 397, 399, 402, 403, 404, 405, 407, 409, 411, 413, 415, 417, 418, 419,
421, 423, 425, 427, 428, 429, 431, 433, 435, 437, 439, 441, 443, 444, 445, 447,
449, 450, 451, 453, 454, 455, 457, 458, 461, 462, 463, 464, 465, 466, 467, 469,
471, 473, 475, 476, 477, 479, 481, 482, 483, 485, 486, 487, 488, 489, 491, 493,
495, 497, 499, 501, 503, 505, 506, 507, 509, 511, 513, 514, 515, 517, 519, 520,
523, 525, 526, 527, 529, 530, 531, 533, 535, 536, 537, 539, 541, 543, 545, 546,
547, 548, 549, 550, 551, 553, 555, 557, 558, 559, 561, 562, 563, 565, 566, 567,
569, 570, 571, 573, 575, 577, 579, 582, 583, 584, 585, 587, 589, 591, 592, 593,
595, 596, 597, 598, 599, 601, 603, 604, 605, 607, 609, 611, 612, 613, 615, 617,
619, 620, 621, 623, 625, 626, 627, 629, 630, 631, 633, 634, 635, 637, 639, 641,
643, 645, 647, 648, 649, 651, 653, 655, 657, 659, 660, 661, 665, 666, 667, 668, 669, 671, 673, 675, 677,
678, 679, 680, 681, 683, 684, 685, 687, 688, 689, 691, 693, 694, 695, 697, 699,
701, 702, 703, 704, 705, 706, 707, 709, 711, 713, 715, 717, 719, 721, 723, 724,
726, 727, 729, 730, 731, 733, 735, 737, 738, 739, 741, 742, 743, 745, 747, 749,
750, 751, 752, 753, 754, 755, 757, 759, 761, 762, 763, 764, 765, 767, 769, 770,
771, 773, 774, 775, 777, 779, 781, 783, 785, 787, 788, 789, 791, 793, 795, 797,
798, 799, 801, 802, 803, 805, 807, 808, 809, 811, 813, 816, 817, 819, 821, 822,
823, 824, 825, 827, 829, 831, 833, 835, 836, 837, 839, 841, 842, 843, 845, 846,
847, 849, 850, 851, 853, 854, 855, 857, 859, 861, 863, 865, 867, 869, 870, 871,
872, 873, 874, 875, 877, 879, 881, 882, 883, 885, 886, 887, 889, 890, 893, 895,
897, 899, 901, 902, 903, 904, 905, 907, 908, 909, 911, 913, 914, 915, 917, 919,
921, 922, 923, 925, 927, 929, 930, 931, 933, 934, 935, 937, 939, 941, 943, 944,
945, 947, 949, 951, 953, 955, 957, 958, 959, 961, 962, 963, 965, 967, 969, 970,
971, 972, 973, 975, 978, 979, 980, 981, 983, 985, 986, 987, 988, 989, 991, 993,
994, 995, 997, 999, 1001, 1003, 1005, 1007, 1009, 1010, 1011, 1013, 1015, 1016,
1017, 1018, 1019, 1021, 1023, 1024, 1025, 1027, 1028, 1029, 1030, 1031, 1033,
1035, 1037, 1039, 1040, 1041, 1043, 1045, 1046, 1047, 1049, 1050, 1051, 1052,
1053, 1055, 1057, 1059, 1060, 1063, 1064, 1065, 1066, 1067, 1069, 1071, 1072,
1073, 1075, 1077, 1079, 1081, 1082, 1083, 1085, 1086, 1087, 1089, 1091, 1092,
1093, 1095, 1097, 1098, 1099, 1101, 1103, 1105, 1107, 1109, 1111, 1113, 1114,
1115, 1117, 1119, 1121, 1123, 1124, 1125, 1126, 1127, 1129, 1131, 1132, 1133,
1135, 1137, 1138, 1139, 1141, 1142, 1143, 1145, 1146, 1147, 1148, 1149, 1151,
1152, 1153, 1154, 1155, 1157, 1158, 1159, 1160, 1161, 1163, 1164, 1165, 1167,
1168, 1169, 1171, 1174, 1175, 1177, 1178, 1179, 1180, 1181, 1183, 1184, 1185,
1187, 1189, 1190, 1191, 1192, 1193, 1195, 1196, 1197, 1199, 1201, 1202, 1203,
1205, 1206, 1207, 1208, 1209, 1210, 1211, 1213, 1215, 1217, 1218, 1219, 1221,
1222, 1223, 1225, 1227, 1229, 1230, 1231, 1233, 1234, 1235, 1237, 1239, 1241,
1243, 1245, 1246, 1247, 1249, 1251, 1253, 1254, 1257, 1258, 1259, 1260, 1261,
1263, 1265, 1266, 1267, 1269, 1271, 1273, 1275, 1276, 1277, 1278, 1279, 1281,
1282, 1283, 1285, 1286, 1287, 1289, 1290, 1291, 1293, 1294, 1295, 1297, 1299,
1301, 1303, 1304, 1305, 1306, 1307, 1309, 1311, 1313, 1315, 1317, 1318, 1319,
1321, 1323, 1325, 1326, 1327, 1329, 1330, 1331, 1332, 1333, 1335, 1337, 1339,
1340, 1341, 1343, 1345, 1347, 1349, 1350, 1351, 1353, 1354, 1355, 1357, 1358,
1359, 1360, 1363, 1365, 1367, 1369, 1370, 1371, 1373, 1375, 1377, 1379, 1381,
1383, 1385, 1387, 1388, 1389, 1391, 1393, 1395, 1397, 1399, 1401, 1403, 1405,
1406, 1407, 1409, 1411, 1412, 1413, 1414, 1415, 1417, 1419, 1421, 1423, 1424,
1425, 1426, 1427, 1429, 1431, 1433, 1434, 1435, 1436, 1437, 1439, 1441, 1442,
1443, 1445, 1447, 1448, 1449, 1451, 1452, 1453, 1455, 1457, 1458, 1459, 1460,
1461, 1462, 1463, 1467, 1469, 1471, 1473, 1474, 1475, 1477, 1479, 1481, 1483,
1485, 1486, 1487, 1489, 1491, 1493, 1495, 1497, 1498, 1499, 1501, 1503, 1505,
1506, 1507, 1509, 1511, 1512, 1513, 1515, 1517, 1518, 1519, 1521, 1522, 1523,
1525, 1527, 1528, 1529, 1531, 1533, 1535, 1537, 1539, 1541, 1543, 1544, 1545,
1547, 1549, 1551, 1552, 1553, 1555, 1557, 1558, 1559, 1561, 1563, 1564, 1565,
1567, 1568, 1569, 1570, 1571, 1573, 1575, 1577, 1578, 1579, 1581, 1585, 1587,
1589, 1591, 1593, 1594, 1595, 1597, 1599, 1600, 1601, 1603, 1604, 1605, 1606,
1607, 1609, 1611, 1613, 1614, 1615, 1616, 1617, 1619, 1621, 1622, 1623, 1625,
1626, 1627, 1628, 1629, 1631, 1632, 1633, 1635, 1637, 1639, 1641, 1642, 1643,
1645, 1646, 1647, 1649, 1651, 1653, 1654, 1655, 1657, 1658, 1659, 1661, 1663,
1664, 1665, 1667, 1669, 1670, 1671, 1673, 1675, 1677, 1679, 1681, 1683, 1686,
1687, 1689, 1691, 1692, 1693, 1695, 1697, 1698, 1699, 1701, 1703, 1705, 1707,
1709, 1710, 1711, 1713, 1714, 1715, 1717, 1718, 1719, 1721, 1723, 1724, 1725,
1726, 1727, 1729, 1731, 1733, 1735, 1737, 1738, 1739, 1741, 1743, 1744, 1745,
1747, 1749, 1750, 1751, 1753, 1754, 1755, 1757, 1759, 1761, 1763, 1765, 1766,
1767, 1769, 1770, 1771, 1772, 1773, 1774, 1775, 1777, 1779, 1781, 1783, 1784,
1785, 1786, 1787, 1789, 1791, 1793, 1795, 1797, 1798, 1799, 1801, 1803, 1804,
1805, 1807, 1809, 1810, 1811, 1813, 1815, 1817, 1818, 1819, 1821, 1823, 1825, 1826,
1828, 1829, 1831, 1833, 1834, 1835, 1837, 1839, 1841, 1842, 1843, 1845, 1847,
1849, 1851, 1853, 1854, 1855, 1857, 1858, 1859, 1861, 1862, 1863, 1864, 1865,
1866, 1867, 1869, 1871, 1873, 1874, 1875, 1877, 1879, 1880, 1881, 1883, 1885,
1886, 1887, 1888, 1889, 1891, 1893, 1894, 1895, 1897, 1899, 1901, 1903, 1904,
1905, 1907, 1909, 1910, 1911, 1913, 1915, 1916, 1917, 1919, 1921, 1923, 1924,
1926, 1927, 1928, 1929, 1930, 1931, 1933, 1935, 1937, 1938, 1939, 1940, 1941,
1942, 1943, 1945, 1947, 1949, 1951, 1953, 1954, 1955, 1957, 1958, 1959, 1960,
1961, 1963, 1965, 1966, 1967, 1969, 1970, 1971, 1972, 1973, 1975, 1976, 1977,
1979, 1981, 1983, 1985, 1986, 1987, 1989, 1991, 1992, 1993, 1995, 1997, 1998,
1999, 2001, 2002, 2003, 2005, 2007, 2008, 2009, 2011, 2013, 2015, 2017, 2019,
2021, 2022, 2023, 2024, 2025, 2027, 2029, 2030, 2031, 2032, 2033, 2034, 2035,
2037, 2039, 2041, 2042, 2043, 2045, 2047, 2048, 2049, 2050, 2051, 2053, 2055,
2057, 2060, 2061, 2063, 2065, 2067, 2069, 2070, 2071, 2072, 2073, 2074, 2075,
2077, 2079, 2081, 2083, 2085, 2087, 2089, 2091, 2092, 2093, 2094, 2095, 2097,
2099, 2101, 2102, 2103, 2105, 2107, 2108, 2109, 2111, 2112, 2113, 2115, 2117,
2119, 2120, 2121, 2123, 2125, 2126, 2127, 2129, 2130, 2131, 2133, 2134, 2135,
2137, 2139, 2141, 2142, 2143, 2145, 2146, 2147, 2149, 2151, 2152, 2153, 2155,
2157, 2159, 2160, 2161, 2163, 2164, 2165, 2167, 2169, 2170, 2171, 2173, 2175,
2177, 2179, 2181, 2182, 2183, 2185, 2186, 2189, 2191, 2193, 2194, 2195, 2197,
2199, 2201, 2203, 2204, 2205, 2207, 2209, 2210, 2211, 2213, 2214, 2215, 2216,
2217, 2219, 2221, 2223, 2224, 2225, 2227, 2229, 2230, 2231, 2233, 2235, 2237,
2238, 2239, 2240, 2241, 2243, 2245, 2247, 2249, 2251, 2252, 2253, 2255, 2257,
2258, 2259, 2261, 2263, 2265, 2266, 2267, 2269, 2271, 2273, 2275, 2276, 2277,
2279, 2281, 2283, 2284, 2285, 2287, 2289, 2290, 2291, 2293, 2295, 2296, 2297,
2299, 2301, 2303, 2305, 2306, 2307, 2309, 2310, 2311, 2313, 2314, 2315, 2316,
2317, 2319, 2320, 2323, 2325, 2327, 2329, 2331, 2333, 2335, 2337, 2339, 2341,
2343, 2345, 2347, 2348, 2349, 2351, 2353, 2355, 2357, 2359, 2360, 2361, 2362,
2363, 2365, 2366, 2367, 2368, 2369, 2370, 2371, 2373, 2374, 2375, 2377, 2379,
2381, 2383, 2384, 2385, 2387, 2389, 2390, 2391, 2393, 2395, 2397, 2398, 2399,
2401, 2403, 2405, 2407, 2409, 2411, 2412, 2413, 2415, 2416, 2417, 2419, 2421,
2422, 2423, 2425, 2427, 2429, 2431, 2433, 2434, 2435, 2437, 2438, 2439, 2440,
2441, 2443, 2444, 2445, 2447, 2449, 2451, 2453, 2454, 2455, 2457, 2461, 2463,
2465, 2466, 2467, 2468, 2469, 2470, 2471, 2473, 2474, 2475, 2477, 2479, 2481,
2482, 2483, 2485, 2487, 2488, 2489, 2491, 2492, 2493, 2495, 2497, 2499, 2501,
2502, 2503, 2504, 2505, 2506, 2507, 2509, 2511, 2512, 2513, 2515, 2516, 2517,
2519, 2521, 2522, 2523, 2525, 2527, 2529, 2531, 2532, 2533, 2534, 2535, 2536,
2537, 2539, 2541, 2543, 2544, 2545, 2547, 2549, 2551, 2552, 2553, 2554, 2555,
2557, 2558, 2559, 2561, 2563, 2565, 2566, 2567, 2569, 2571, 2572, 2573, 2575,
2577, 2579, 2581, 2582, 2583, 2585, 2587, 2588, 2589, 2591, 2593, 2594, 2595,
2597, 2599, 2600, 2601, 2603, 2605, 2606, 2607, 2609, 2610, 2611, 2612, 2613,
2614, 2615, 2619, 2621, 2623, 2624, 2625, 2627, 2629, 2631, 2632, 2633, 2635,
2637, 2639, 2641, 2642, 2643, 2645, 2646, 2647, 2649, 2650, 2651, 2652, 2653,
2655, 2657, 2659, 2660, 2661, 2663, 2664, 2665, 2667, 2669, 2671, 2673, 2674,
2675, 2677, 2679, 2681, 2683, 2685, 2686, 2687, 2689, 2691, 2693, 2695, 2697,
2699, 2700, 2701, 2703, 2704, 2705, 2707, 2708, 2709, 2710, 2711, 2713, 2715,
2717, 2719, 2720, 2721, 2722, 2723, 2725, 2726, 2727, 2728, 2729, 2731, 2733,
2734, 2735, 2737, 2738, 2739, 2742, 2743, 2744, 2745, 2747, 2749, 2750, 2751,
2752, 2753, 2754, 2755, 2757, 2759, 2761, 2763, 2765, 2766, 2767, 2769, 2771,
2772, 2773, 2775, 2777, 2779, 2780, 2781, 2783, 2785, 2787, 2789, 2791, 2792,
2793, 2794, 2795, 2797, 2799, 2801, 2803, 2805, 2807, 2809, 2811, 2812, 2813,
2815, 2816, 2817, 2818, 2819, 2821, 2823, 2825, 2827, 2828, 2829, 2831, 2833,
2835, 2837, 2839, 2841, 2842, 2843, 2844, 2845, 2847, 2849, 2850, 2851, 2853,
2854, 2855, 2857, 2859, 2861, 2863, 2864, 2865, 2866, 2867, 2869, 2871, 2873,
2874, 2875, 2877, 2879, 2881, 2882, 2883, 2884, 2885, 2887, 2888, 2889, 2890,
2891, 2893, 2895, 2898, 2899, 2901, 2902, 2903, 2905, 2906, 2907, 2909, 2911,
2913, 2914, 2915, 2917, 2919, 2921, 2923, 2924, 2925, 2927, 2928, 2929, 2931,
2933, 2935, 2937, 2938, 2939, 2941, 2943, 2945, 2946, 2947, 2948, 2949, 2951,
2952, 2953, 2954, 2955, 2957, 2959, 2961, 2962, 2963, 2965, 2967, 2969, 2971,
2972, 2973, 2974, 2975, 2977, 2978, 2979, 2981, 2982, 2983, 2984, 2985, 2987, 2989,
2991, 2992, 2993, 2994, 2995, 2997, 2999, 3001, 3002, 3003, 3005, 3007, 3009,
3010, 3011, 3013, 3014, 3015, 3016, 3017, 3019, 3021, 3023, 3025, 3026, 3027,
3029, 3031, 3033, 3034, 3035, 3036, 3037, 3039, 3040, 3041, 3042, 3043, 3045,
3046, 3047, 3049, 3050, 3053, 3054, 3055, 3057, 3059, 3061, 3063, 3065, 3066,
3067, 3068, 3069, 3071, 3072, 3073, 3075, 3077, 3079, 3081, 3082, 3083, 3085,
3087, 3089, 3090, 3091, 3093, 3094, 3095, 3097, 3099, 3101, 3103, 3105, 3106,
3107, 3109, 3111, 3113, 3115, 3116, 3117, 3119, 3121, 3122, 3123, 3125, 3127,
3128, 3129, 3131, 3133, 3135, 3137, 3138, 3139, 3141, 3143, 3145, 3146, 3147,
3149, 3150, 3151, 3153, 3155, 3157, 3159, 3161, 3162, 3163, 3165, 3166, 3167,
3169, 3171, 3173, 3175, 3176, 3177, 3178, 3179, 3181, 3183, 3184, 3185, 3187,
3188, 3189, 3190, 3191, 3193, 3195, 3197, 3199, 3200, 3201, 3202, 3203, 3205,
3207, 3208, 3209, 3211, 3212, 3213, 3215, 3217, 3219, 3222, 3223, 3224, 3225,
3226, 3227, 3229, 3230, 3231, 3233, 3234, 3235, 3237, 3239, 3241, 3243, 3245,
3247, 3249, 3251, 3252, 3253, 3254, 3255, 3257, 3259, 3261, 3262, 3263, 3265,
3267, 3269, 3271, 3272, 3273, 3274, 3275, 3277, 3279, 3280, 3281, 3283, 3284,
3285, 3287, 3289, 3291, 3292, 3293, 3294, 3295, 3297, 3299, 3301, 3303, 3304,
3305, 3307, 3308, 3309, 3310, 3311, 3313, 3315, 3317, 3318, 3319, 3321, 3322,
3323, 3325, 3327, 3329, 3330, 3331, 3333, 3334, 3335, 3337, 3338, 3339, 3341,
3343, 3345, 3347, 3349, 3351, 3352, 3353, 3354, 3355, 3357, 3359, 3361, 3363,
3366, 3367, 3368, 3369, 3371, 3372, 3373, 3375, 3376, 3377, 3379, 3381, 3383,
3385, 3387, 3389, 3390, 3391, 3393, 3394, 3395, 3397, 3399, 3401, 3403, 3405,
3406, 3407, 3408, 3409, 3411, 3413, 3415, 3417, 3419, 3421, 3422, 3423, 3425,
3427, 3429, 3431, 3433, 3435, 3436, 3437, 3439, 3441, 3442, 3443, 3445, 3446,
3447, 3449, 3450, 3451, 3452, 3453, 3455, 3457, 3459, 3461, 3463, 3464, 3465,
3467, 3469, 3471, 3472, 3473, 3475, 3476, 3477, 3478, 3479, 3481, 3483, 3485,
3486, 3487, 3489, 3490, 3491, 3493, 3495, 3497, 3499, 3500, 3501, 3502, 3503,
3505, 3507, 3509, 3510, 3511, 3513, 3514, 3515, 3517, 3519, 3521, 3522, 3523,
3525, 3527, 3529, 3531, 3532, 3533, 3535, 3537, 3538, 3539, 3540, 3541, 3543,
3545, 3546, 3547, 3548, 3549, 3551, 3553, 3554, 3557, 3558, 3559, 3560, 3561,
3563, 3565, 3567, 3569, 3570, 3571, 3573, 3574, 3575, 3577, 3578, 3579, 3581,
3582, 3583, 3585, 3587, 3589, 3591, 3592, 3593, 3595, 3596, 3597, 3599, 3600,
3601, 3603, 3605, 3606, 3607, 3608, 3609, 3611, 3613, 3614, 3615, 3617, 3619,
3621, 3622, 3623, 3625, 3627, 3629, 3630, 3631, 3633, 3634, 3635, 3637, 3639,
3641, 3643, 3644, 3645, 3647, 3649, 3650, 3651, 3652, 3653, 3654, 3655, 3657,
3658, 3659, 3661, 3663, 3665, 3667, 3668, 3669, 3670, 3671, 3673, 3675, 3677,
3679, 3681, 3683, 3684, 3685, 3687, 3688, 3689, 3691, 3693, 3695, 3697, 3698,
3699, 3703, 3704, 3705, 3707, 3708, 3709, 3711, 3713, 3714, 3715, 3716, 3717,
3719, 3721, 3722, 3723, 3725, 3727, 3729, 3730, 3731, 3732, 3733, 3735, 3737,
3738, 3739, 3741, 3743, 3745, 3747, 3749, 3750, 3751, 3753, 3754, 3755, 3757,
3758, 3759, 3761, 3762, 3763, 3765, 3766, 3767, 3769, 3771, 3772, 3773, 3775,
3777, 3779, 3781, 3783, 3784, 3785, 3787, 3789, 3791, 3793, 3795, 3797, 3799,
3800, 3801, 3802, 3803, 3805, 3806, 3807, 3809, 3811, 3813, 3814, 3815, 3817,
3819, 3820, 3821, 3823, 3824, 3825, 3826, 3827, 3829, 3830, 3831, 3833, 3835,
3836, 3837, 3839, 3840, 3841, 3843, 3845, 3847, 3848, 3849, 3851, 3852, 3853,
3854, 3855, 3856, 3857, 3859, 3861, 3863, 3865, 3866, 3867, 3869, 3870, 3871,
3873, 3874, 3875, 3876, 3877, 3879, 3880, 3881, 3883, 3884, 3885, 3889, 3891,
3893, 3895, 3896, 3897, 3898, 3899, 3901, 3903, 3905, 3906, 3907, 3908, 3909,
3910, 3911, 3913, 3914, 3915, 3917, 3919, 3920, 3921, 3923, 3925, 3927, 3928,
3929, 3931, 3932, 3933, 3935, 3937, 3939, 3940, 3941, 3943, 3945, 3946, 3947,
3949, 3951, 3952, 3953, 3954, 3955, 3957, 3959, 3961, 3962, 3963, 3965, 3967,
3969, 3971, 3972, 3973, 3975, 3977, 3978, 3979, 3981, 3983, 3984, 3985, 3987,
3989, 3991, 3993, 3994, 3995, 3997, 3999, 4001, 4002, 4003, 4004, 4005, 4007,
4009, 4011, 4013, 4015, 4016, 4017, 4018, 4019, 4021, 4022, 4023, 4025, 4026,
4027, 4028, 4029, 4031, 4032, 4033, 4035, 4037, 4039, 4041, 4043, 4044, 4045,
4047, 4048, 4049, 4050, 4051, 4053, 4054, 4055, 4057, 4058, 4059, 4061, 4062,
4063, 4067, 4069, 4071, 4072, 4073, 4075, 4077, 4079, 4081, 4083, 4084, 4085,
4087, 4089, 4090, 4091, 4093, 4095, 4096, 4097, 4099, 4101, 4103, 4104, 4105,
4107, 4109, 4110, 4111, 4113, 4114, 4115, 4117, 4119, 4121, 4123, 4124, 4125,
4126, 4127, 4129, 4130, 4131, 4133, 4134, 4135, 4137, 4139, 4141, 4143, 4144, 4145,
4147, 4149, 4151, 4152, 4153, 4154, 4155, 4157, 4159, 4161, 4162, 4163, 4165,
4166, 4167, 4169, 4170, 4171, 4173, 4175, 4177, 4179, 4181, 4182, 4183, 4185,
4186, 4187, 4188, 4189, 4191, 4192, 4193, 4195, 4197, 4198, 4199, 4201, 4203,
4204, 4205, 4207, 4209, 4211, 4212, 4213, 4215, 4217, 4218, 4219, 4221, 4222,
4223, 4225, 4227, 4229, 4230, 4231, 4233, 4234, 4235, 4237, 4239, 4240, 4243,
4245, 4246, 4247, 4249, 4251, 4253, 4254, 4255, 4257, 4259, 4261, 4262, 4263,
4265, 4267, 4268, 4269, 4270, 4271, 4272, 4273, 4275, 4277, 4279, 4281, 4282,
4283, 4285, 4287, 4289, 4290, 4291, 4293, 4294, 4295, 4296, 4297, 4299, 4301,
4303, 4304, 4305, 4307, 4309, 4310, 4311, 4313, 4315, 4317, 4319, 4320, 4321,
4323, 4325, 4326, 4327, 4328, 4329, 4331, 4333, 4335, 4336, 4337, 4339, 4340,
4341, 4342, 4343, 4345, 4347, 4348, 4349, 4351, 4353, 4354, 4355, 4357, 4359,
4361, 4363, 4364, 4365, 4367, 4369, 4371, 4373, 4375, 4377, 4379, 4380, 4381,
4383, 4384, 4385, 4387, 4388, 4389, 4390, 4391, 4393, 4395, 4397, 4399, 4401,
4403, 4405, 4407, 4409, 4411, 4413, 4414, 4415, 4417, 4418, 4419, 4422, 4423,
4425, 4426, 4427, 4429, 4431, 4432, 4433, 4435, 4437, 4439, 4441, 4443, 4445,
4446, 4447, 4448, 4449, 4450, 4451, 4453, 4454, 4455, 4457, 4459, 4460, 4461,
4463, 4465, 4467, 4469, 4470, 4471, 4473, 4475, 4477, 4479, 4481, 4482, 4483,
4485, 4486, 4487, 4489, 4491, 4493, 4495, 4497, 4498, 4499, 4500, 4501, 4503,
4505, 4506, 4507, 4509, 4511, 4513, 4514, 4515, 4516, 4517, 4519, 4521, 4522,
4523, 4525, 4526, 4527, 4528, 4529, 4531, 4533, 4534, 4535, 4537, 4539, 4541,
4542, 4543, 4544, 4545, 4547, 4549, 4550, 4551, 4552, 4553, 4554, 4555, 4557,
4559, 4560, 4561, 4563, 4565, 4567, 4568, 4569, 4571, 4573, 4575, 4577, 4578,
4579, 4580, 4581, 4582, 4583, 4585, 4587, 4589, 4590, 4591, 4593, 4594, 4595,
4597, 4598, 4599, 4600, 4601, 4602, 4603, 4605, 4607, 4609, 4611, 4613, 4615,
4617, 4619, 4621, 4623, 4625, 4626, 4627, 4628, 4629, 4630, 4631, 4633, 4637,
4639, 4640, 4641, 4643, 4645, 4647, 4648, 4649, 4651, 4653, 4654, 4655, 4657,
4659, 4661, 4663, 4664, 4665, 4666, 4667, 4669, 4670, 4671, 4673, 4675, 4677,
4678, 4679, 4681, 4683, 4685, 4686, 4687, 4689, 4690, 4691, 4692, 4693, 4695,
4697, 4699, 4700, 4701, 4703, 4705, 4706, 4707, 4709, 4711, 4713, 4714, 4715,
4717, 4719, 4720, 4721, 4723, 4725, 4727, 4729, 4731, 4732, 4733, 4735, 4737,
4739, 4741, 4743, 4745, 4746, 4747, 4748, 4749, 4751, 4753, 4755, 4757, 4759,
4760, 4761, 4762, 4763, 4765, 4767, 4769, 4770, 4771, 4773, 4774, 4775, 4777,
4779, 4781, 4783, 4785, 4787, 4789, 4790, 4791, 4792, 4793, 4795, 4796, 4797,
4799, 4801, 4803, 4805, 4806, 4809, 4810, 4811, 4813, 4815, 4816, 4817, 4819,
4821, 4822, 4823, 4825, 4827, 4829, 4831, 4833, 4834, 4835, 4837, 4838, 4839,
4840, 4841, 4843, 4844, 4845, 4847, 4849, 4850, 4851, 4853, 4854, 4855, 4857,
4859, 4860, 4861, 4863, 4865, 4866, 4867, 4868, 4869, 4871, 4873, 4875, 4877,
4878, 4879, 4881, 4882, 4883, 4885, 4886, 4887, 4888, 4889, 4891, 4892, 4893,
4894, 4895, 4897, 4899, 4901, 4903, 4904, 4905, 4906, 4907, 4908, 4909, 4911,
4912, 4913, 4915, 4917, 4919, 4921, 4922, 4923, 4924, 4925, 4927, 4929, 4931,
4932, 4933, 4934, 4935, 4937, 4939, 4940, 4941, 4943, 4945, 4947, 4949, 4950,
4951, 4953, 4954, 4955, 4957, 4959, 4961, 4963, 4964, 4965, 4967, 4969, 4971,
4973, 4974, 4975, 4976, 4977, 4979, 4981, 4983, 4985, 4987, 4988, 4989, 4990,
4991, 4993, 4995, 4997, 4999, ...
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