This site is supported by donations to The OEIS Foundation.
User:Eric Angelini/FirstDiffPrimes
The first differences of S are the prime numbers of S
[English version below]
On met en gras les nombres premiers de la suite S ; la succession des nombres gras 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, 150, 287, 24, 175, 36, 185, 348, 17, 184, ...
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, 131, 127, 263, 151, 139, 149, 163, 331, 167, ...
Si la quantité de nombres gras pour l’instant est si faible, (quatre seulement) 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).
__________
In bold characters are the prime numbers of the sequence S; the bold terms are also the absolute first differences of S, appearing in the same order:
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, ...
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, 131, 127, 263, 151, 139, 149, 163, 331, 167, ...
The quantity of bold terms so far in S is low (only four until now); this is 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.
__________
[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,...
__________
[December 12th, 2010]
Alois Heinz has just sent me the sequence SP with the same idea -- but dealing with Semi-Primes:
SP = 1, 5, 11, 2, 12, 27, 13, 47, 8, 29, 3, 28, 50, 4, 37, 72, 7, 45, 96, 19, 68, 6, 61, 130, 9, 66, 124, 18, 92, 10, 97, 182, 16, 102, 193, 15, 108, 14, 109, 220, 17, 132, 250, 24, 147, 266, 31, 153, 20, 149, 283, 30, 171, 313, 48, 191, 32, 190, 44, 189, 34, 195, 364, 23, 200, 383, 54, 239, 52, 246, 40, 241, 36, 238, 452, 39, 248, 461, 63, 280, 21, 236, 455, 60, 281, 499, 26, 263, 510, 25, 279, 528, 59, 333, 42, 304, 571, 56, 343, 22, 300, 595, 41, 330, 628, 75, 374, 71, 372, 46, 348, 43, 352, 33, 347, 670, 35, 369, 696, 67, 402, 741, 70, 416, 774, 53, 408, 769, 80, 442, 65, 430, 801, 38, 419, 805, 88, 470, 76, 467, 64, 457, 864, 51, 462, 877, 78, 495, 73, 500, 937, 99, 544, 90, 536, 83, 530, 79, 550, 84, 577, 1035, 77, 555, 1036, 81, 563, 49, 546, 1048, 89, 578, 1083, 98, 599, 62, 588, 55, 572, 1091, 112, 639, 101, 612, 69, 598, 1140, 103, 638, 1183, 104, 663, 1225, 110, 661, 1240, 105, 678, 113, 702, 116, 682, 1265, 100, 691, 1272, 125, 722, 1336, 117, 728, 1350, 107, 730, 1356, 151, 784, 135, 790, 121, 783, 1450, 57, 736, 1370, 136, 810, 1491, 120, 814, 127, 812, 114, 809, 1506, 137, 836, 58, 761, 1467, 128, 846, 139, 852, 106, 829, 74, 808, 1539, 82, 819, 1564, 87, 840, 1598, 85, 834, 1600, 131, 902, 1669, 140, 919, 126, 907, 1692, 154, 945, 138, 927, 1721, 144, 946, 1749, 160, 975, 157, 988, 1805, 150, 991, 148, 983, 1825, 152, 1001, 1852, 163, 1025, 156, 1021, 1887, 170, 1049, 1920, 166, 1044, 1930, 164, 1053, 1946, 165, 1060, 86, 984, 1885, 167, 1066, 1971, 168, 1085, 162, 1075, 1989, 172, 1093, 2015, 174, 1100, 2034, 91, 1024, 1973, 176, 1127, 184, 1123, 2084, 175, 1148, 2113, 179, 1161, 2150, 173, 1166, 2161, 178, 1176, 2179, 188, 1194, 93, 1104, 2111, 192, 1210, 2237, 186, 1228, 181, 1222, 2265, 94, 1151, 2197, 95, 1150, 2223, 197, 1256, 111, 1178, 2255, 196, 1277, 2371, 180, 1279, 2390, 204, 1323, 2405, 199, 1320, 2457, 208, 1334, 2448, 203, 1341, 115, 1248, 2387, 198, 1340, 2494, 211, 1352, 2511, 232, 1381, 118, 1275, 2442, 223, 1392, 2583, 222, 1396, 207, 1384, 2570, 228, 1423, 212, 1410, 2609, 240, 1442, 2645, 210, 1424, 2662, 216, 1443, 2650, 224, 1471, 230, 1483, 226, 1481, 2742, 225, 1487, 2754, 123, 1394, 2667, 233, 1515, 229, 1523, 2808, 227, 1520, 119, 1425, 2724, 235, 1548, 231, 1549, 234, 1556, 2889, 251, 1580, 243, 1586, 2932, 271, 1616, 261, 1608, 122, 1476, 2825, 252, 1609, 258, 1621, 242, 1624, 2990, 244, 1627, 3012, 255, 1644, 257, 1648, 245, 1647, 133, 1530, 2945, 264, 1675, 129, 1534, 2952, 134, 1551, 2988, 253, 1694, 256, 1710, 141, 1602, 3067, 260, 1733, 142, 1620, 3121, 275, 1741, 3243, 265, 1772, 3289, 272, 1794, 143, 1640, 3167, 268, 1777, 159, 1696, 3231, 158, 1699, 3260, 269, 1824, 146, 1709, 3283, 270, 1855, 290, 1884, 145, 1748, 3370, 155, 1778, 3411, 277, 1908, 3547, 161, 1802, 3444, 286, 1932, 169, 1812, 3461, 288, 1949, 3607, 273, 1944, 3598, 284, 1965, 3644, 293, 1978, 3681, 294, 2000, 3691, 276, 1987, 3674, 297, 2004, 3717, 296, 2010, 3736, 317, 2044, 282, 2013, 3748, 306, 2057, 292, 2037, 3794, 307, 2068, 3842, 324, 2093, 3872, 285, 2080, 3879, 310, 2124, 331, 2152, 315, 2134, 312, 2141, 3948, 337, 2172, 4010, 316, 2169, 311, 2154, 3993, 328, 2185, 320, 2202, 308, 2203, 354, 2205, 322, 2196, 4089, 340, 2243, 4134, 325, 2222, 4128, 341, 2256, 4177, 318, 2241, 4168, 177, 2106, 4047, 332, 2269, 4208, 342, 2284, 183, 2128, 4082, 336, 2293, 4252, 349, 2310, 329, 2292, 4259, 338, 2304, 185, 2170, 4139, 376, 2358, 353, 2347, 4365, 187, 2208, 4239, 194, 2213, 4246, 344, 2382, 4423, 345, 2392, 350, 2399, 4444, 357, 2423, 206, 2268, 4339, 201, 2274, 4352, 366, 2464, 359, 2436, 205, 2300, 4403, 360, 2477, 351, 2473, 202, 2325, 4452, 389, 2527, 368, 2523, 356, 2503, 4652, 367, 2524, 4689, 370, 2543, 214, 2388, 4565, 378, 2560, 379, 2562, 363, 2557, 4752, 373, 2574, 4783, 384, 2613, 4828, 375, 2593, 4820, 385, 2619, 4865, 387, 2640, 4898, 396, 2659, 4916, 405, 2690, 388, 2679, 4984, 390, 2696, 5003, 410, 2725, 399, 2716, 397, 2718, 5077, 392, 2715, 380, 2707, 5060, 401, 2763, 400, 2774, 5160, 406, 2797, 5192, 414, 2816, 409, 2822, 5265, 412, 2861, 428, 2847, 418, 2871, 413, 2840, 5295, 209, 2670, 5132, 421, 2884, 5355, 440, 2909, 426, 2900, 5379, 424, 2905, 404, 2891, 5382, 213, 2708, 5205, 434, 2943, 398, 2896, 5409, 420, 2927, 5442, 425, 2944, 5477, 432, 2950, 217, 2775, 5312, 259, 2820, 215, 2769, 5328, 431, 2994, 423, 2989, 5556, 429, 3006, 219, 2788, 5370, 436, 3014, 5608, 395, 2982, 221, 2810, 5412, 433, 3032, 5635, 444, 3050, 435, 3049, 218, 2829, 5452, 498, 3125, 456, 3105, 463, 3092, 438, 3079, 5730, 443, 3096, 5797, 448, 3169, 464, 3186, 439, 3082, 5805, 460, 3203, 441, 3175, 5934, 449, 3220, 6018, 473, 3246, 6025, 476, 3285, 472, 3257, 6068, 459, 3282, 6097, 468, 3307, 480, 3321, 6175, 237, 3100, 5945, 483, 3342, 247, 3102, 5960, 487, 3354, 485, 3363, 450, 3316, 254, 3135, 6020, 249, 3150, 6043, 465, 3359, 6265, 488, 3390, 469, 3380, 274, 3192, 6121, 484, 3407, 474, 3405, 6340, 291, 3232, 6174, 475, 3422, 6373, 518, 3467, 262, 3221, 6183, 506, 3471, 490, 3456, 267, 3240, 6214, 491, 3468, 6451, 486, 3472, 477, 3464, 6462, 479, 3484, 6477, 508, 3515, 6544, 503, 3525, 494, 3529, 6556, 523, 3562, 509, 3552, 6599, 492, 3538, 6601, 516, 3573, 496, 3567, 6632, 513, 3604, 507, 3592, 6678, 512, 3605, 504, 3617, 6715, 522, 3621, 6724, 515, 3632, 525, 3643, 6770, 287, 3420, 6551, 520, 3659, 321, 3463, 6606, 524, 3671, 6820, 278, 3429, ...
And here is agraph showing the sequence SP (red) and the first differences (semi-primes in blue):
Thanks to all contrubutors! Best, É.
