Primes and prime sums

On Mon, Jan 14, 2013 at 1:19 PM, Eric Angelini wrote:

Hello Seq-Fans,

Write down the prime numbers:  2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Insert between two primes the smallest integer not yet present in the sequence such that two neighboring integers sum to a prime.

I think the sequence starts:  2,0,3,8,5,6,7,12,11,18,13,24,17, ...

Best,

É.

________________________

[Maximilian Hasler] :

Can anyone find the index at which will appear the number 4?

[Jack Brennen] :

It won’t.  If you have p,4,q in the sequence, that would mean that:

p, q, p+4, and q+4 are all prime.

Assuming p>3, if q = p+2, then p+4 is divisible by 3.

Assuming p>3, if q = p+4, then q+4 is divisible by 3.

Assuming p>3, if q > p+4, then p+4 is not prime.

[Hans Havermann] :

All even numbers (except 4) will get used. Soon enough they will always be smaller than their adjacent primes. Not as obvious is that they will follow two distinct trajectories. Why?

[Richard Guy] :

Will any numbers 6n+4 be used?

[Hans Havermann] :

The first 100 terms are: {2, 0, 3, 8, 5, 6, 7, 12, 11, 18, 13, 24, 17, 42, 19, 48, 23, 14, 29, 30, 31, *10, 37, 60, 41, 66, 43, 36, 47, 20, 53, 44, 59, 78, 61, *22, 67, 96, 71, 108, 73, *28, 79, 84, 83, 68, 89, 102, 97, 126, 101, 90, 103, 120, 107, 72, 109, 54, 113, 114, 127, 150, 131, 26, 137, 132, 139, 144, 149, 162, 151, *16, 157, *34, 163, 186, 167, 56, 173, 50, 179, 168, 181, 156, 191, 228, 193, 204, 197, 222, 199, *40, 211, *46, 223, 174, 227, 192, 229, 210}. The asterisked terms are 6n+4.

[Maximilian Hasler] :

Among the first 5000 even terms of the sequence, the smallest missing ones are 4 and 2*[2707,2789,2855,2858,2864,2876,2879,2888,2894,2918,2927,2942,2948,2954,2963,2969,2987,2992,2999,...]

Only 2 of these are 6n+2, all others 6n+4.

All of these are used later on. I computed the sequence further, and all even numbers > 4 *at least* up to approx. 15 000 are used.

so I’m rather confident that all even terms (except 4) are there; but the terms 6n+4 come later than the 6n+2:

Some more data for the records:

The even numbers which "momentarily" are the smallest unused numbers (among those > 4) are:

6, 10, 16, 32, 184, 268, 334, 388, 482, 544, 604, 628, 632, 788, 928, 958, 1052, 1076, 1186, 1202, 1214, 1312, 1316, 1318, 1384, 1432, 1532, 1552, 1594, 1852, 1958, 2182, 2236, 2248, 2312, 2416, 2440, 2488, 2542, 2572, 2612, 2644, 2836, 2906, 2936, 3626, 3628, 3704, 3772, 3868, 3988, 4252, 4258, 4336, 4366, 4390, 4412, 4438, 4442, 4444, 4594, 4672, 4792, 4832, 4882, 5074, 5218, 5242, 5356, 5398, 5414, 5578, 5728, 5854, 6278, 6418, 6632, 7082, 7142, 7318, 7618, 7622, 7628, 7684, 7702, 7768, 7858, 8068, 8212, 8692, 8776, 8956, 9122, 9238, 9244, 9292, 9304, 9424, 9448, 9784, 9806, 9868, 9872, 9992, 10018, 10084, 10162, 10348, 10552, 10768, 10834, 11008, 11422, 11552, 11618, 11702, 11738, 11752, 11930, 11936, 11974, 12094, 12482, 12538, 12568, 13234, 13582, 13666, 13894

[Jean-Marc Falcoz] :

J’ai fait un petit programme pour visualiser cette suite. Au sujet de ces trajectoires, les premiers sont tous de la forme 6k+1 ou 6k+5. Quand on choisit deux premiers consécutifs, on a un peu moins d’une chance sur deux pour qu’ils soient tous deux de la forme 6k+1 ou tous deux de la forme 6k+5 (et donc dans une légère majorité de cas, on a le cas mixte : 6k+1 et 6k+5). Si l’on s’occupe de ce cas (6k+1 et 6k+5) on ne peut bien sûr pas trouver de nombre pair de la forme 6n+2 ou 6n+4 qui donnerait deux premiers par addition, on doit forcément prendre un multiple de 6. On a donc une majorité de cas pour lesquels on doit choisir une minorité de nombres (un multiple de 6 = 33% des nombres pairs), cela explique que ces multiples de 6 progressent bien plus vite dans la suite que ceux de la forme 6n+2 ou 6n+4 et que les trajectoires soient si marquées.

Pour le graphe, la trajectoire supérieure est celle des nombres premiers, la trajectoire médiane est celle des multiples de 6, et celle du bas est celle des 6n+2 et 6n+4.

Le problème dual est sympa aussi (voir ci-dessous après les termes de la suite d’origine) :

{2, 0, 3, 8, 5, 6, 7, 12, 11, 18, 13, 24, 17, 42, 19, 48, 23, 14, 29, 30, 31, 10, 37, 60, 41, 66, 43, 36, 47, 20, 53, 44, 59, 78, 61, 22, 67, 96, 71, 108, 73, 28, 79, 84, 83, 68, 89, 102, 97, 126, 101, 90, 103, 120, 107, 72, 109, 54, 113, 114, 127, 150, 131, 26, 137, 132, 139, 144, 149, 162, 151, 16, 157, 34, 163, 186, 167, 56, 173, 50, 179, 168, 181, 156, 191, 228, 193, 204, 197, 222, 199, 40, 211, 46, 223, 174, 227, 192, 229, 210, 233, 38, 239, 180, 241, 138, 251, 80, 257, 74, 263, 104, 269, 252, 271, 76, 277, 336, 281, 240, 283, 216, 293, 264, 307, 306, 311, 258, 313, 300, 317, 246, 331, 52, 337, 270, 347, 294, 349, 324, 353, 188, 359, 234, 367, 94, 373, 70, 379, 360, 383, 158, 389, 312, 397, 276, 401, 198, 409, 282, 419, 390, 421, 288, 431, 378, 433, 124, 439, 318, 443, 98, 449, 372, 457, 366, 461, 348, 463, 396, 467, 134, 479, 342, 487, 420, 491, 330, 499, 354, 503, 110, 509, 92, 521, 498, 523, 118, 541, 100, 547, 462, 557, 170, 563, 164, 569, 450, 571, 82, 577, 432, 587, 140, 593, 230, 599, 492, 601, 220, 607, 214, 613, 474, 617, 402, 619, 142, 631, 408, 641, 510, 643, 444, 647, 86, 653, 200, 659, 570, 661, 136, 673, 414, 677, 146, 683, 426, 691, 480, 701, 522, 709, 384, 719, 504, 727, 130, 733, 88, 739, 540, 743, 486, 751, 106, 757, 456, 761, 468, 769, 528, 773, 516, 787, 612, 797, 212, 809, 618, 811, 438, 821, 606, 823, 600, 827, 624, 829, 594, 839, 714, 853, 576, 857, 630, 859, 564, 863, 546, 877, 552, 881, 726, 883, 666, 887, 636, 907, 582, 911, 648, 919, 678, 929, 642, 937, 660, 941, 122, 947, 116, 953, 654, 967, 696, 971, 62, 977, 254, 983, 588, 991, 112, 997, 154, 1009, 558, 1013, 218, 1019, 702, 1021, 756, 1031, 690, 1033, 58, 1039, 684, 1049, 672, 1051, 750, 1061, 810, 1063, 160, 1069, 190, 1087, 780, 1091, 786, 1093, 774, 1097, 194, 1103, 128, 1109, 762, 1117, 64, 1123, 178, 1129, 738, 1151, 720, 1153, 708, 1163, 816, 1171, 822, 1181, 242, 1187, 236, 1193, 888, 1201, 226, 1213, 870, 1217, 206, 1223, 224, 1229, 768, 1231, 196, 1237, 202, 1249, 744, 1259, 272, 1277, 804, 1279, 858, 1283, 260, 1289, 792, 1291, 262, 1297, 840, 1301, 828, 1303, 834, 1307, 152, 1319, 918, 1321, 166, 1327, 852, 1361, 182, 1367, 176, 1373, 900, 1381, 172, 1399, 732, 1409, 798, 1423, 846, 1427, 882, 1429, 864, 1433, 308, 1439, 894, 1447, 930, 1451, 1098, 1453, 148, 1459, 238, 1471, 876, 1481, 1068, 1483, 906, 1487, 1062, 1489, 948, 1493, 248, 1499, 32, 1511, 266, 1523, 936, 1531, 280, 1543, 328, 1549, 534, 1553, 314, 1559, 972, 1567, 1050, 1571, 960, 1579, 1038, 1583, 996, 1597, 942, 1601, 386, 1607, 984, 1609, 1008, 1613, 374, 1619, 1092, 1621, 250, 1627, 912, 1637, 1020, 1657, 244, 1663, 954, 1667, 924, 1669, 208, 1693, 966, 1697, 990, 1699, 978, 1709, 278, 1721, 1080, 1723, 1044, 1733, 1056, 1741, 232, 1747, 316, 1753, 310, 1759, 304, 1777, 286, 1783, 1014, 1787, 1002, 1789, 298, 1801, 1032, 1811, 320, 1823, 1086, 1831, 1122, 1847, 1110, 1861, 346, 1867, 1152, 1871, 1128, 1873, 1146, 1877, 1242, 1879, 1074, 1889, 350, 1901, 380, 1907, 434, 1913, 290, 1931, 1188, 1933, 1134, 1949, 1170, 1951, 1116, 1973, 368, 1979, 1230, 1987, 364, 1993, 1026, 1997, 1254, 1999, 1164, 2003, 1206, 2011, 256, 2017, 1296, 2027, 1140, 2029, 1284, 2039, 1218, 2053, 1260, 2063, 404, 2069, 302, 2081, 1176, 2083, 1104, 2087, 1212, 2089, 1224, 2099, 440, 2111, 1248, 2113, 1194, 2129, 1200, 2131, 412, 2137, 1182, 2141, 1158, 2143, 1314, 2153, 1308, 2161, 370, 2179, 430, 2203, 1326, 2207, 344, 2213, 1236, 2221, 1290, 2237, 1302, 2239, 1374, 2243, 1380, 2251, 1266, 2267, 1272, 2269, 1344, 2273, 1350, 2281, 376, 2287, 400, 2293, 1320, 2297, 362, 2309, 1362, 2311, 1386, 2333, 338, 2339, 1332, 2341, 352, 2347, 1446, 2351, 326, 2357, 1422, 2371, 322, 2377, 1416, 2381, 1440, 2383, 274, 2389, 1404, 2393, 284, 2399, 392, 2411, 296, 2417, 644, 2423, 1356, 2437, 1410, 2441, 356, 2447, 590, 2459, 1452, 2467, 436, 2473, 1434, 2477, 1524, 2503, 340, 2521, 1398, 2531, 1392, 2539, 1338, 2543, 518, 2549, 1368, 2551, 406, 2557, 1470, 2579, 458, 2591, 1428, 2593, 1464, 2609, 1482, 2617, 1512, 2621, 416, 2633, 1506, 2647, 1554, 2657, 1500, 2659, 1494, 2663, 1548, 2671, 292, 2677, 526, 2683, 1764, 2687, 1530, 2689, 1758, 2693, 470, 2699, 1590, 2707, 1740, 2711, 1518, 2713, 184, 2719, 1542, 2729, 1488, 2731, 1596, 2741, 1278, 2749, 1698, 2753, 1476, 2767, 1560, 2777, 332, 2789, 1632, 2791, 460, 2797, 1650, 2801, 1458, 2803, 1638, 2819, 1674, 2833, 1614, 2837, 464, 2843, 1800, 2851, 472, 2857, 1656, 2861, 452, 2879, 1704, 2887, 1620, 2897, 410, 2903, 398, 2909, 1734, 2917, 1566, 2927, 530, 2939, 1608, 2953, 1686, 2957, 500, 2963, 494, 2969, 1578, 2971, 1680, 2999, 1722, 3001, 1662, 3011, 1572, 3019, 1710, 3023, 1626, 3037, 1602, 3041, 1752, 3049, 478, 3061, 466, 3067, 382, 3079, 1854, 3083, 428, 3089, 1842, 3109, 1824, 3119, 1668, 3121, 1536, 3137, 1794, 3163, 1770, 3167, 1932, 3169, 358, 3181, 490, 3187, 1746, 3191, 506, 3203, 488, 3209, 1584, 3217, 1716, 3221, 1782, 3229, 1830, 3251, 1848, 3253, 1914, 3257, 1974, 3259, 268, 3271, 1788, 3299, 1980, 3301, 496, 3307, 454, 3313, 448, 3319, 1908, 3323, 524, 3329, 1692, 3331, 520, 3343, 1884, 3347, 560, 3359, 1872, 3361, 1818, 3371, 1728, 3373, 2130, 3389, 1890, 3391, 2112, 3407, 614, 3413, 1776, 3433, 2070, 3449, 1950, 3457, 1956, 3461, 2016, 3463, 2040, 3467, 1812, 3469, 2010, 3491, 2028, 3499, 418, 3511, 502, 3517, 1806, 3527, 1992, 3529, 1998, 3533, 620, 3539, 1878, 3541, 532, 3547, 1860, 3557, 1920, 3559, 442, 3571, 1836, 3581, 1896, 3583, 1644, 3593, 2046, 3607, 394, 3613, 1866, 3617, 476, 3623, 2118, 3631, 580, 3637, 574, 3643, 2058, 3659, 572, 3671, 1968, 3673, 2064, 3677, 1962, 3691, 562, 3697, 2082, 3701, 1938, 3709, 2142, 3719, 2022, 3727, 484, 3733, 550, 3739, 2052, 3761, 596, 3767, 2100, 3769, 2088, 3779, 2034, 3793, 1944, 3797, 710, 3803, 536, 3821, 2268, 3823, 2214, 3833, 2106, 3847, 2076, 3851, 2238, 3853, 1986, 3863, 2004, 3877, 1902, 3881, 2148, 3889, 784, 3907, 2136, 3911, 446, 3917, 2172, 3919, 2124, 3923, 584, 3929, 2160, 3931, 706, 3943, 2256, 3947, 2154, 3967, 2184, 3989, 602, 4001, 2196, 4003, 2466, 4007, 554, 4013, 548, 4019, 2178, 4021, 616, 4027, 2094, 4049, 2220, 4051, 586, 4057, 2190, 4073, 578, 4079, 512, 4091, 2208, 4093, 424, 4099, 538, 4111, 1926, 4127, 2232, 4129, 2340, 4133, 650, 4139, 2424, 4153, 2316, 4157, 2202, 4159, 514, 4177, 742, 4201, 2280, 4211, 692, 4217, 2334, 4219, 2262, 4229, 2322, 4231, 2250, 4241, 2310, 4243, 2328, 4253, 698, 4259, 2292, 4261, 2418, 4271, 2298, 4273, 2406, 4283, 668, 4289, 2274, 4297, 592, 4327, 2226, 4337, 2352, 4339, 2370, 4349, 2304, 4357, 610, 4363, 2346, 4373, 686, 4391, 716, 4397, 704, 4409, 758, 4421, 2358, 4423, 598, 4441, 640, 4447, 2286, 4451, 542, 4457, 656, 4463, 638, 4481, 2388, 4483, 2478, 4493, 2166, 4507, 724, 4513, 2394, 4517, 2244, 4519, 2448, 4523, 800, 4547, 2400, 4549, 712, 4561, 670, 4567, 2430, 4583, 2376, 4591, 556, 4597, 334, 4603, 568, 4621, 2556, 4637, 2490, 4639, 2568, 4643, 764, 4649, 2562, 4651, 622, 4657, 646, 4663, 2514, 4673, 734, 4679, 422, 4691, 728, 4703, 860, 4721, 2586, 4723, 508, 4729, 2748, 4733, 680, 4751, 2460, 4759, 388, 4783, 2694, 4787, 2520, 4789, 2454, 4793, 770, 4799, 2412, 4801, 718, 4813, 2664, 4817, 2706, 4831, 808, 4861, 2382, 4871, 566, 4877, 812, 4889, 2634, 4903, 748, 4909, 2580, 4919, 722, 4931, 2526, 4933, 2544, 4937, 626, 4943, 2508, 4951, 700, 4957, 2364, 4967, 2592, 4969, 2604, 4973, 2550, 4987, 664, 4993, 658, 4999, 2538, 5003, 740, 5009, 2868, 5011, 2496, 5021, 2436, 5023, 2484, 5039, 608, 5051, 2532, 5059, 634, 5077, 2610, 5081, 482, 5087, 752, 5099, 2658, 5101, 760, 5107, 544, 5113, 688, 5119, 2442, 5147, 674, 5153, 2574, 5167, 2472, 5171, 2502, 5179, 2688, 5189, 2730, 5197, 604, 5209, 652, 5227, 2646, 5231, 2718, 5233, 2640, 5237, 806, 5261, 818, 5273, 794, 5279, 2598, 5281, 2652, 5297, 776, 5303, 908, 5309, 2628, 5323, 2736, 5333, 2616, 5347, 2742, 5351, 782, 5381, 830, 5387, 824, 5393, 878, 5399, 2682, 5407, 790, 5413, 2676, 5417, 2670, 5419, 682, 5431, 766, 5437, 2796, 5441, 2778, 5443, 754, 5449, 2760, 5471, 866, 5477, 2754, 5479, 2790, 5483, 746, 5501, 2886, 5503, 2766, 5507, 854, 5519, 2700, 5521, 676, 5527, 2892, 5531, 2712, 5557, 796, 5563, 694, 5569, 2724, 5573, 2850, 5581, 2772, 5591, 2838, 5623, 2808, 5639, 2898, 5641, 922, 5647, 2622, 5651, 2946, 5653, 2970, 5657, 2880, 5659, 2904, 5669, 2844, 5683, 628, 5689, 2934, 5693, 2820, 5701, 2826, 5711, 662, 5717, 2784, 5737, 2802, 5741, 2856, 5743, 820, 5749, 772, 5779, 2910, 5783, 2916, 5791, 2940, 5801, 872, 5807, 896, 5813, 3018, 5821, 832, 5827, 862, 5839, 2964, 5843, 980, 5849, 2988, 5851, 802, 5857, 2832, 5861, 842, 5867, 2952, 5869, 2862, 5879, 2958, 5881, 2922, 5897, 926, 5903, 3048, 5923, 2814, 5927, 902, 5939, 3060, 5953, 3156, 5981, 1010, 5987, 3042, 6007, 3000, 6011, 938, 6029, 3012, 6037, 826, 6043, 3090, 6047, 914, 6053, 2976, 6067, 844, 6073, 838, 6079, 3072, 6089, 3150, 6091, 3036, 6101, 848, 6113, 2928, 6121, 3006, 6131, 3108, 6133, 2994, 6143, 3030, 6151, 850, 6163, 3120, 6173, 1040, 6197, 3084, 6199, 3078, 6203, 3138, 6211, 892, 6217, 2982, 6221, 3192, 6229, 730, 6247, 3174, 6257, 950, 6263, 944, 6269, 3162, 6271, 916, 6277, 2874, 6287, 920, 6299, 3132, 6301, 3102, 6311, 1100, 6317, 890, 6323, 884, 6329, 3210, 6337, 910, 6343, 3168, 6353, 1058, 6359, 3270, 6361, 886, 6367, 736, 6373, 814, 6379, 3024, 6389, 3234, 6397, 1036, 6421, 1030, 6427, 3294, 6449, 3180, 6451, 778, 6469, 3318, 6473, 3066, 6481, 3198, 6491, 968, 6521, 3282, 6529, 904, 6547, 3240, 6551, 3126, 6553, 3186, 6563, 1118, 6569, 3288, 6571, 880, 6577, 3252, 6581, 992, 6599, 3204, 6607, 940, 6619, 970, 6637, 3096, 6653, 1028, 6659, 3348, 6661, 856, 6673, 868, 6679, 3054, 6689, 3378, 6691, 3402, 6701, 3228, 6703, 874, 6709, 3222, 6719, 3360, 6733, 3366, 6737, 632, 6761, 3246, 6763, 3144, 6779, 3258, 6781, 3312, 6791, 3216, 6793, 3276, 6803, 3336, 6823, 3354, 6827, 3264, 6829, 3330, 6833, 3300, 6841, 3432, 6857, 1016, 6863, 1004, 6869, 3462, 6871, 946, 6883, 3390, 6899, 3414, 6907, 3426, 6911, 956, 6917, 1046, 6947, 3324, 6949, 3372, 6959, 3342, 6961, 1120, 6967, 3486, 6971, 1082, 6977, 1034, 6983, 3408, 6991, 1090, 6997, 3456, 7001, 1088, 7013, 998, 7019, 3114, 7027, 1042, 7039, 3420, 7043, 3444, 7057, 1012, 7069, 3384, 7079, 1130, 7103, 1160, 7109, 1070, 7121, 932, 7127, 3582, 7129, 3438, 7151, 3480, 7159, 934, 7177, 3546, 7187, 974, 7193, 3306, 7207, 3522, 7211, 3498, 7213, 898, 7219, 3504, 7229, 3552, 7237, 994, 7243, 3840, 7247, 1064, 7253, 1136, 7283, 3564, 7297, 3474, 7307, 3630, 7309, 1192, 7321, 3450, 7331, 3528, 7333, 3534, 7349, 3510, 7351, 1018, 7369, 1054, 7393, 976, 7411, 952, 7417, 3540, 7433, 836, 7451, 1172, 7457, 3600, 7459, 1150, 7477, 3606, 7481, 1142, 7487, 3570, 7489, 3558, 7499, 3744, 7507, 3576, 7517, 1106, 7523, 788, 7529, 3714, 7537, 3636, 7541, 1166, 7547, 3612, 7549, 3702, 7559, 3792, 7561, 1108, 7573, 3666, 7577, 1154, 7583, 1124, 7589, 3468, 7591, 1006, 7603, 3654, 7607, 3762, 7621, 988, 7639, 3618, 7643, 1244, 7649, 3774, 7669, 3588, 7673, 3648, 7681, 982, 7687, 3396, 7691, 3798, 7699, 3768, 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__________

[Éric Angelini, later on SeqFan Mailing List]:

Hello Seq-Fans,

Write down the non prime numbers: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...

Insert between two non-primes the smallest prime not yet present in the sequence such that two neighboring integers sum to a non-prime.

I think the sequence starts: 1,5,4,2,6,19,8,7,9,11,10,23,12,13,14,31,15,17,16,...

Best,

É.

__________

[Jean-Marc Falcoz] :

Salut Eric, tu as raison, le problème dual est sympa aussi. Voici la suite que j’obtiens (si quelqu’un pouvait vérifier les termes, ce serait bien, on ne sait jamais...)

Au sujet des illustrations, je me suis d’abord dit que ça devait être faux en regardant la seconde, car la trajectoire du haut (les nombres premiers) ne devrait pas donner une droite, mais en posant une règle sur l’écran (quel bricolage !) on voit que ce n’est pas une droite du tout, heureusement.

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Merci Jean-Marc – and thanks to all contributors!

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