The first differences of S

are the prime numbers of S

[English version below]

On jaunit les nombres premiers de la suite S ; la suite des nombres jaunis est aussi celle des premières différences de S :

S = 1, 4, 6, 25, 8, 3, 10, 21, 34, 57, 14, 45, 16, 63, 2, 39, 80, 9, 62, 121, 12, 85, 18, 115, 26, 105, 22, 123, 20, 133, 240, 49, 176, 19...

D =   3  2 19  17  5  7  11  13  23  43, 41, 29, 47, 61,37, 41, 71,53, 59, 109, 73, 67, 97,  89, 79,  83, 101, 103,113, 107, 191,127, 157...

Si la quantité de nombres jaunis pour l’instant est si faible, c’est que :

- S doit être aussi une permutation de N ;

- S doit être la première suite lexicographique répondant aux contraintes ;

- D doit être une permutation de P (P est la suite des nombres premiers).

Qui pourrait vérifier le début de S ci-dessus et m’aider à calculer un millier de termes ?

à+

É.

__________

We color in yellow the prime numbers of the sequence S; the yellow terms are also the absolute first differences of S:

S = 1, 4, 6, 25, 8, 3, 10, 21, 34, 57, 14, 45, 16, 63, 2, 39, 80, 9, 62, 121, 12, 85, 18, 115, 26, 105, 22, 123, 20, 133, 240, 49, 176, 19...

D =   3  2 19  17  5  7  11  13  23  43, 41, 29, 47, 61,37, 41, 71,53, 59, 109, 73, 67, 97,  89, 79,  83, 101, 103,113, 107, 191,127, 157...

The quantity of yellow terms so far in S is low because:

- we want S to be a permutation of N;

- we want S to be the lexicographically first sequence obeying all constraints;

- we want D to be a permutation of the prime numbers.

Could someone please check the above terms and help me computing thousand more or so?

Best,

É.

__________

[December 8th, 2010]

Douglas McNeil, David Newman and Alois Heinz were quick to answer (and to correct my hand-computed try above):

S = 1, 4, 6, 25, 8, 3, 10, 21, 34, 57, 14, 45, 16, 63, 2, 39, 80, 9, 62, 121, 12, 85, 18, 115, 26, 105, 22, 123, 20, 133, 240, 49, 176, 19, 150, 287, 24, 175, 36, 185, 348, 17, 184, 5, 178, 371, 54, 235, 38, 237, 448, 15, 238, 465, 44, 273, 32, 265, 504, 55, 306, 35, 292, 561, 40, 321, 28, 305, 588, 65, 372, 685, 42, 391, 702, 95, 432, 779, 46, 399, 758, 7, 374, 747, 56, 435, 52, 441, 838, 11, 412, 831, 58, 489, 50, 459, 902, 81, 538, 51, 512, 13, 476, 943, 60, 539, 30, 533, 1024, 27, 568, 1115, 64, 621, 1184, 33, 602, 1173, 70, 657, 1234, 111, 704, 87, 686, 1287, 74, 687, 68, 699, 1340, 91, 738, 77, 730, 1389, 82, 755, 72, 749, 48, 767, 1476, 23, 750, 1507, 78, 817, 1560, 161, 922, 69, 856, 1625, 66, 875, 1672, 75, 886, 1715, 88, 927, 104, 961, 84, 965, 102, 989, 130, 1037, 90, 1001, 1920, 43, 972, 31, 968, 1921, 98, 1065, 94, 1071, 2054, 141, 1132, 93, 1102, 2115, 76, 1095, 2116, 29, 1060, 2093, 96, 1145, 2206, 117, 1180, 2249, 106, 1197, 100, 1209, 92, 1179, 86, 1215, 2368, 99, 1262, 2433, 122, 1309, 108, 1325, 132, 1313, 2536, 119, 1348, 47, 1278, 2515, 116, 1375, 2652, 61, 1344, 2623, 120, 1411, 114, 1403, 2706, 155, 1474, 147, 1468, 2829, 110, 1477, 2850, 143, 1524, 37, 1446, 2869, 128, 1555, 2988, 145, 1584, 41, 1488, 2947, 144, 1595, 112, 1593, 3064, 125, 1614, 3107, 136, 1635, 124, 1647, 3178, 129, 1678, 71, 1624, 53, 1620, 3199, 138, 1739, 118, 1701, 3310, 59, 1696, 3309, 140, 1797, 134, 1803, 3422, 109, 1776, 3473, 126, 1819, 3518, 159, 1868, 135, 1856, 73, 1796, 3537, 146, 1893, 3646, 177, 1936, 3713, 142, 1929, 3718, 171, 1972, 3783, 152, 1983, 3830, 153, 2014, 3885, 158, 2025, 3898, 165, 2044, 3933, 154, 2055, 148, 2079, 4012, 189, 2138, 187, 2160, 4147, 180, 2159, 156, 2149, 4148, 201, 2212, 183, 2200, 4227, 170, 2223, 160, 2229, 4310, 67, 2150, 4249, 198, 2327, 166, 2277, 164, 2295, 4432, 203, 2344, 4497, 200, 2379, 172, 2375, 162, 2401, 4622, 213, 2450, 169, 2412, 4685, 168, 2419, 4686, 205, 2492, 195, 2488, 4797, 194, 2527, 186, 2525, 174, 2545, 188, 2535, 4912, 209, 2590, 207, 2596, 4989, 190, 2601, 5024, 231, 2668, 221, 2662, 5121, 202, 2669, 192, 2665, 5186, 217, 2748, 5287, 206, 2755, 212, 2769, 5348, 247, 2840, 219, 2828, 97, 2714, 5361, 182, 2815, 5472, 245, 2904, 215, 2878, 89, 2760, 5447, 210, 2893, 216, 2915, 204, 2933, 220, 2913, 5662, 225, 2978, 5745, 214, 2991, 5782, 255, 3052, 5853, 196, 3015, 5848, 267, 3104, 243, 3094, 5951, 208, 3087, 5974, 285, 3182, 279, 3188, 79, 2996, 5949, 232, 3159, 6116, 249, 3212, 6181, 228, 3227, 226, 3237, 218, 3241, 6278, 291, 3332, 253, 3320, 6403, 230, 3349, 260, 3381, 244, 3411, 224, 3387, 6568, 297, 3488, 259, 3462, 6671, 222, 3439, 6660, 83, 3336, 6593, 234, 3493, 6764, 295, 3594, 275, 3576, 6905, 246, 3553, 6876, 323, 3654, 7015, 236, 3579, 6950, 261, 3634, 101, 3490, 6897, 278, 3711, 248, 3661, 7110, 319, 3776, 103, 3564, 7031, 250, 3741, 242, 3753, 7270, 299, 3826, 7355, 252, 3791, 7332, 113, 3670, 7251, 254, 3813, 7396, 327, 3920, 303, 3910, 107, 3720, 7343, 264, 3901, 258, 3949, 290, 3961, 7658, 289, 3990, 7709, 276, 3985, 7724, 375, 4136, 315, 4082, 7851, 262, 4055, 7852, 329, 4162, 309, 4156, 8007, 266, 4143, 280, 4161, 272, 4179, 256, 4167, 8084, 325, 4244, 301, 4230, 8249, 286, 4275, 268, 4199, 8200, 191, 4194, 8207, 270, 4291, 8318, 127, 4176, 8255, 304, 4377, 284, 4375, 8474, 157, 4268, 8395, 294, 4427, 274, 4403, 8542, 333, 4490, 8649, 296, 4473, 8674, 131, 4342, 8559, 316, 4535, 282, 4541, 300, 4531, 8792, 345, 4616, 343, 4626, 8915, 288, 4615, 8952, 355, 4694, 9051, 298, 4647, 9010, 341, 4714, 9105, 302, 4699, 9120, 407, 4830, 9271, 308, 4755, 9206, 339, 4796, 9259, 318, 4811, 328, 4835, 312, 4825, 9344, 393, 4940, 357, 4906, 9503, 322, 4883, 9450, 437, 5028, 365, 4986, 335, 4972, 9611, 330, 4979, 336, 5015, 324, 4981, 9654, 361, 5082, 9805, 314, 5043, 310, 5061, 9820, 381, 5164, 351, 5138, 9927, 326, 5127, 9944, 405, 5236, 137, 4998, 9869, 358, 5235, 332, 5221, 10130, 363, 5294, 10227, 320, 5257, 10200, 413, 5364, 377, 5334, 10307, 334, 5301, 10294, 411, 5410, 263, 5266, 10275, 344, 5355, 10376, 369, 5392, 10431, 338, 5389, 10448, 387, 5464, 10551, 340, 5453, 346, 5465, 10618, 395, 5562, 10751, 352, 5523, 10720, 417, 5626, 10857, 356, 5589, 10850, 423, 5696, 415, 5694, 385, 5682, 10985, 354, 5677, 11010, 151, 5498, 10849, 350, 5731, 11118, 139, 5532, 10931, 342, 5755, 11162, 429, 5846, 403, 5822, 11253, 362, 5803, 11252, 453, 5924, 445, 5922, 11405, 378, 5885, 364, 5865, 11368, 485, 6004, 447, 6010, 149, 5718, 11291, 382, 5973, 11596, 425, 6064, 11705, 376, 6023, 370, 6021, 11680, 483, 6152, 451, 6134, 11845, 398, 6135, 386, 6127, 434, 6213, 392, 6175, 368, 6159, 11960, 469, 6282, 455, 6294, 163, 6006, 11855, 366, 6223, 12074, 331, 6200, 12079, 360, 6241, 12138, 167, 6070, 11993, 396, 6335, 408, 6415, 404, 6385, 12414, 427, 6464, 12507, 388, 6435, 12488, 477, 6544, 471, 6550, 12639, 400, 6501, 380, 6471, 12584, 475, 6608, 12751, 422, 6573, 410, 6609, 406, 6603, 12814, 513, 6730, 179, 6400, 12629, 402, 6649, 12906, 473, 6736, 173, 6442, 12719, 430, 6717, 394, 6693, 12994, 497, 6808, 13125, 428, 6757, 384, 6721, 13064, 481, 6834, 13195, 414, 6811, 390, 6769, 13136, 495, 6884, 193, 6620, 13071, 418, 6891, 13372, 519, 7010, 13531, 438, 6985, 416, 6945, 13516, 507, 7088, 13695, 436, 7035, 13672, 501, 7154, 493, 7166, 13845, 424, 7125, 13828, 515, 7224, 317, 7036, 13769, 442, 7179, 13940, 181, 6944, 13737, 440, 7263, 460, 7287, 446, 7275, 14108, 511, 7368, 505, 7374, 197, 7068, 13951, 452, 7363, 464, 7381,...

Below is David’s Mathematica program; many thanks to all contributors!

(the equivalent sequence where “prime” is replaced by “non-primes” is here); the same sequence, due to Alois Heinz but dealing with semi-primes, is there)

__________

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