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*