Primes/non-primes alternate

(+ digits)

- a non-prime integer must be followed by a prime;

- a prime must be followed by a non-prime;

- the same with the sequence's digits (non-prime digits are 0,1,4,6,8,9);

- start with a(1) = 1

- take always the smallest available integer not already present in the sequence and not leading to a contradiction - olé!

T = 1, 2, 4, 3, 6, 5, 8, 7, 9, 29, 20, 31, 21, 59, 24, 71, 26, 79, 28, 263, 12, 13, 15, 17...

(...)

[this is a nightmare to compute by hand; please forgive the possible errors above]

Best,

É.

[Maximilian Hasler]:

A217560(n, show_all=0, a=1, u)={for( i=2, n, u+=1<<a; show_all & print1(a", "); for(t=1, 9e9, bittest(u, t) & next; isprime(t)==bittest(i, 0) & next; isprime(a%10)==isprime(t\10^(#Str(t)-1)) & (t+=10^(#Str(t)-1)-1) & next; my(tt=t); while( tt>9, isprime(tt%10)==isprime((tt\=10)%10) & next(2)); a=t; break)); a}

This is now in the OEIS: https://oeis.org/A217560

A simplified variant: https://oeis.org/A217559

[Jean-Marc Falcoz] (see the graph at the end of the page):

T = 1, 2, 4, 3, 6, 5, 8, 7, 9, 29, 20, 31, 21, 59, 24, 71, 26, 79, 28, 263, 12, 13, 15, 17, 42, 43, 45, 47, 62, 67, 63, 83, 65, 97, 82, 131, 30, 283, 85, 139, 34, 293, 87, 151, 36, 307, 92, 179, 38, 313, 93, 421, 39, 317, 95, 431, 50, 347, 120, 367, 121, 383, 124, 397, 126, 503, 128, 547, 129, 563, 130, 587, 134, 593, 136, 743, 138, 787, 150, 797, 154, 2029, 51, 2039, 54, 2129, 56, 2131, 58, 2179, 70, 2459, 74, 2621, 76, 2659, 78, 2671, 202, 439, 203, 479, 205, 631, 207, 659, 212, 821, 213, 829, 215, 839, 217, 859, 242, 929, 243, 971, 245, 1213, 156, 2851, 247, 1217, 158, 2879, 262, 1283, 159, 2939, 265, 1297, 170, 2971, 267, 1303, 171, 3079, 282, 1307, 174, 3121, 285, 1367, 176, 3631, 287, 1543, 178, 3659, 292, 1567, 420, 3671, 295, 1583, 424, 3821, 297, 1597, 426, 3851, 302, 1747, 428, 3929, 303, 1783, 429, 3931, 305, 1787, 430, 5021, 312, 4217, 434, 5039, 315, 4243, 436, 5051, 342, 4283, 438, 5059, 343, 4297, 450, 5171, 345, 4363, 451, 5179, 362, 4397, 454, 5431, 363, 4507, 456, 5471, 365, 4513, 458, 5479, 382, 4517, 459, 5639, 385, 4547, 470, 5651, 387, 4567, 471, 5659, 392, 4583, 474, 5821, 393, 4597, 476, 5839, 395, 4703, 478, 5851, 502, 4783, 620, 5879, 505, 4787, 621, 5939, 507, 4793, 624, 7039, 512, 6203, 626, 7079, 513, 6217, 628, 7121, 515, 6247, 629, 7129, 517, 6263, 630, 7151, 542, 6287, 634, 7159, 543, 6317, 636, 7451, 545, 6343, 638, 7459, 562, 6367, 639, 7621, 565, 6397, 650, 7639, 567, 6547, 651, 7829, 582, 6563, 654, 7879, 583, 6703, 656, 7951, 585, 6763, 658, 20287, 670, 20297, 671, 20347, 674, 20393, 676, 20507, 678, 20543, 679, 20563, 820, 20593, 824, 20707, 826, 20717, 828, 20743, 830, 20747, 831, 21247, 834, 21283, 836, 21313, 838, 21317, 850, 21347, 851, 21383, 854, 21397, 856, 21503, 858, 21517, 870, 21563, 871, 21587, 874, 21713, 876, 21767, 878, 21787, 879, 24203, 920, 24247, 921, 24317, 924, 24517, 926, 24547, 928, 24593, 930, 24763, 931, 24767, 934, 24793, 936, 26203, 938, 26263, 939, 26267, 950, 26293, 951, 26297, 954, 26317, 956, 26347, 958, 26387, 959, 26393, 970, 26513, 974, 26597, 976, 26713, 978, 26717, 979, 26783, 1202, 6793, 1203, 8243, 1205, 8263, 1207, 8287, 1212, 8293, 1215, 8297, 1242, 8317, 1243, 8363, 1245, 8387, 1247, 8513, 1262, 8543, 1263, 8563, 1265, 8597, 1267, 8707, 1282, 8713, 1285, 8747, 1287, 8783, 1292, 9203, 1293, 9283, 1295, 9293, 1302, 9343, 1305, 9397, 1312, 9547, 1313, 9587, 1315, 9743, 1317, 9767, 1342, 9787, 1343, 12071, 592, 12421, 595, 12451, 597, 12479, 702, 12659, 703, 12671, 705, 12821, 707, 12829, 712, 12959, 713, 12979, 715, 13121, 717, 13151, 742, 13159, 745, 13171, 747, 13421, 762, 13451, 763, 13679, 765, 13829, 767, 13831, 782, 13859, 783, 13879, 785, 13921, 792, 13931, 793, 15031, 795, 15121, 2020, 28283, 1345, 15131, 2021, 28297, 1347, 15139, 2024, 28307, 1362, 15439, 2026, 28387, 1363, 15451, 2028, 28393, 1365, 15629, 2030, 28513, 1382, 15671, 2031, 28517, 1383, 15679, 2034, 28547, 1385, 15859, 2036, 28597, 1387, 15959, 2038, 28703, 1392, 15971, 2050, 28793, 1393, 17021, 2051, 29207, 1395, 17029, 2054, 29243, 1397, 17159, 2056, 29287, 1502, 17431, 2058, 29297, 1503, 17471, 2059, 29303, 1505, 17659, 2070, 29347, 1507, 17839, 2071, 29363, 1512, 17851, 2074, 29383, 1513, 17921, 2076, 29387, 1515, 17929, 2078, 29567, 1517, 17939, 2079, 29587, 1542, 17959, 2120, 29717, 1545, 17971, 2121, 30203, 1547, 42071, 2124, 30293, 1562, 42131, 2126, 30307, 1563, 42139, 2128, 30313, 1565, 42179, 2130, 30347, 1582, 42451, 2134, 30367, 1585, 42821, 2136, 30517, 1587, 42829, 2138, 30593, 1592, 42839, 2139, 30703, 1593, 42859, 2150, 30707, 1595, 42929, 2151, 30713, 1702, 42979, 2154, 30763, 1703, 43051, 2156, 31247, 1705, 43151, 2158, 31267, 1707, 43159, 2159, 31307, 1712, 43451, 2170, 31387, 1713, 43651, 2171, 31393, 1715, 43951, 2174, 31397, 1717, 45121, 2176, 31513, 1742, 45131, 2178, 31517, 1743, 45139, 2420, 31543, 1745, 45179, 2421, 31547, 1762, 45439, 2424, 31567, 1763, 45631, 2426, 31583, 1765, 45659, 2428, 31793, 1767, 45821, 2429, 34213, 1782, 45959, 2430, 34217, 1785, 45971, 2431, 34267, 1792, 45979, 2434, 34283, 1793, 47051, 2436, 34297, 1795, 47059, 2438, 34303, 1797, 47129, 2439, 34313, 4202, 47431, 2450, 34367, 4203, 47459, 2451, 34513, 4205, 47629, 2454, 34543, 4207, 47639, 2456, 34583, 4212, 47659, 2458, 34703, 4213, 47939, 2470, 34747, 4215, 47951, 2471, 34763, 4242, 62039, 2474, 36217, 4245, 62071, 2476, 36263, 4247, 62129, 2478, 36293, 4262, 62131, 2479, 36307, 4263, 62171, 2620, 36313, 4265, 62459, 2624, 36343, 4267, 62639, 2626, 36383, 4282, 62659, 2628, 36563, 4285, 62851, 2629, 36583, 4287, 62921, 2630, 36587, 4292, 62929, 2631, 36713, 4293, 62939, 2634, 36767, 4295, 62971, 2636, 36787, 4302, 63029, 2638, 36793, 4303, 63031, 2639, 38287, 4305, 63059, 2650, 38303, 4307, 63079, 2651, 38317, 4312, 63131, 2654, 38393, 4313, 63179, 2656, 38543, 4315, 63421, 2658, 38567, 4317, 63439, 2670, 38593, 4342, 63629, 2674, 38707, 4343, 63659, 2676, 38713, 4345, 63671, 2678, 38747, 4347, 63839, 2679, 38767, 4362, 63929, 2820, 38783, 4365, 65029, 2821, 39217, 4367, 65071, 2824, 39293, 4382, 65129, 2826, 39313, 4383, 65171, 2828, 39317, 4385, 65179, 2829, 39343, 4387, 65479, 2830, 39367, 4392, 65629, 2831, 39383, 4393, 65651, 2834, 39397, 4395, 65831, 2836, 39503, 4502, 65839, 2838, 39563, 4503, 65851, 2839, 39703, 4505, 65921, 2850, 50207, 4512, 65929, 2854, 50263, 4515, 65951, 2856, 50287, 4542, 67021, 2858, 50363, 4543, 67079, 2859, 50383, 4545, 67121, 2870, 50387, 4562, 67129, 2871, 50503, 4563, 67139, 2874, 50513, 4565, 67421, 2876, 50543, 4582, 67429, 2878, 50587, 4585, 67631, 2920, 50593, 4587, 67651, 2921, 50707, 4592, 67679, 2924, 50767, 4593, 67829, 2926, 51203, 4595, 67931, 2928, 51217, 4702, 67939, 2929, 51263, 4705, 67979, 2930, 51283, 4707, 82021, 2931, 51287, 4712, 82031, 2934, 51307, 4713, 82039, 2936, 51343, 4715, 82051, 2938, 51347, 4717, 82129, 2950, 51383, 4742, 82139, 2951, 51503, 4743, 82171, 2954, 51517, 4745, 82421, 2956, 51563, 4747, 82471, 2958, 51593, 4762, 82651, 2959, 51713, 4763, 82939, 2970, 51767, 4765, 83059, 2974, 51787, 4767, 83071, 2976, 51797, 4782, 83431, 2978, 54217, 4785, 83459, 2979, 54287, 4792, 83471, 3020, 54293, 4795, 83621, 3021, 54347, 4797, 83639, 3024, 54367, 6202, 83921, 3026, 54503, 6205, 83939, 3028, 54517, 6207, 85021, 3029, 54547, 6212, 85121, 3030, 54563, 6213, 85159, 3031, 54583, 6215, 85429, 3034, 54713, 6242, 85439, 3036, 54767, 6243, 85451, 3038, 54787, 6245, 85621, 3039, 56207, 6262, 85639, 3050, 56263, 6265, 85829, 3051, 56267, 6267, 85831, 3054, 56383, 6282, 85931, 3056, 56393, 6283, 87071, 3058, 56503, 6285, 87121, 3059, 56543, 6292, 87151, 3070, 56597, 6293, 87179, 3071, 56713, 6295, 87421, 3074, 56747, 6297, 87629, 3076, 56767, 6302, 87631, 3078, 56783, 6303, 87671, 3120, 58207, 6305, 87679, 3124, 58217, 6307, 87931, 3126, 58243, 6312, 87959, 3128, 58313, 6313, 92051, 3129, 58363, 6315, 92179, 3130, 58367, 6342, 92431, 3131, 58393, 6345, 92459, 3134, 58543, 6347, 92479, 3136, 58567, 6362, 92639, 3138, 58763, 6363, 92671, 3139, 58787, 6365, 92821, 3150, 59207, 6382, 92831, 3151, 59243, 6383, 92921, 3154, 59263, 6385, 92951, 3156, 59387, 6387, 92959, 3158, 59393, 6392, 93059, 3159, 59513, 6393, 93131, 3170, 59567, 6395, 93139, 3171, 59707, 6502, 93151, 3174, 59743, 6503, 93179, 3176, 59747, 6505, 93479, 3178, 59797, 6507, 93629, 3179, 70207, 6512, 93851, 3420, 70297, 6513, 93871, 3421, 70313, 6515, 93971, 3424, 70393, 6517, 93979, 3426, 70507, 6542, 95021, 3428, 70583, 6543, 95071, 3429, 70717, 6545, 95131, 3430, 70783, 6562, 95429, 3431, 70793, 6565, 95471, 3434, 71263, 6567, 95479, 3436, 71287, 6582, 95621, 3438, 71293, 6583, 95629, 3439, 71317, 6585, 95651, 3450, 71347, 6587, 95929, 3451, 71363, 6592, 95959, 3454, 71387, 6593, 95971, 3456, 71503, 6595, 97021, 3458, 71563, 6597, 97039, 3459, 71593, 6702, 97151, 3470, 71597, 6705, 97159, 3471, 71707, 6707, 97171, 3474, 71713, 6712, 97429, 3476, 74203, 6713, 97459, 3478, 74287, 6715, 97651, 3479, 74293, 6717, 97829, 3620, 74297, 6742, 97859, 3621, 74317, 6743, 97871, 3624, 74363, 6745, 97879, 3626, 74383, 6747, 97931, 3628, 74507, 6762, 120247, 6765, 120283, 6767, 120293, 6782, 120383, 6783, 120397, 6785, 120503, 6787, 120563, 6792, 120587, 6795, 120713, 6797, 120763, 8202, 120767, 8203, 121267, 8205, 121283, 8207, 121313, 8212, 121343, 8213, 121367, 8215, 121507, 8217, 121547, 8242, 121763, 8245, 121787, 8247, 124213, 8262, 124247, 8265, 124297, 8267, 124303, 8282, 124343, 8283, 124363, 8285, 124367, 8292, 124513, 8295, 124543, 8302, 124567, 8303, 124703, 8305, 124717, 8307, 124783, 8312, 124793, 8313, 126307, 8315, 126317, 8342, 126397, 8343, 126517, 8345, 126547, 8347, 126583, 8362, 126703, 8365, 126713, 8367, 126743, 8382, 128203, 8383, 128213, 8385, 128287, 8392, 128347, 8393, 128393, 8395, 128563, 8397, 128717, 8502, 128747, 8503, 128767, 8505, 129263, 8507, 129287, 8512, 129293, 8515, 129313, 8517, 129347, 8542, 129517, 8545, 129587, 8547, 129593, 8562, 129707, 8565, 129763, 8567, 129793, 8582, 130267, 8583, 130303, 8585, 130307, 8587, 130343, 8592, 130363, 8593, 130367, 8595, 130513, 8702, 130517, 8703, 130547, 8705, 130783, 8712, 130787, 8715, 131203, 8717, 131213, 8742, 131267, 8743, 131293, 8745, 131297, 8762, 131303, 8763, 131317, 8765, 131363, 8767, 131507, 8782, 131543, 8785, 131707, 8787, 131713, 8792, 131743, 8793, 131783, 8795, 131797, 8797, 134207, 9202, 134213, 9205, 134243, 9207, 134263, 9212, 134287, 9213, 134293, 9215, 134363, 9217, 134503, 9242, 134507, 9243, 134513, 9245, 134587, 9247, 134593, 9262, 134597, 9263, 134707, 9265, 136207, 9267, 136217, 9282, 136247, 9285, 136303, 9287, 136343, 9292, 136393, 9295, 136397, 9297, 136547, 9302, 138247, 9303, 138283, 9305, 138517, 9307, 138547, 9312, 138563, 9313, 138587, 9315, 138763, 9317, 138793, 9342, 138797, 9345, 139267, 9347, 139297, 9362, 139303, 9363, 139313, 9365, 139343, 9367, 139367, 9382, 139387, 9383, 139393, 9385, 139397, 9387, 139547, 9392, 139597, 9393, 139703, 9395, 139747, 9502, 139787, 9503, 150203, 9505, 150217, 9507, 150247, 9512, 150287, 9513, 150343, 9515, 150383, 9517, 150503, 9542, 150517, 9543, 150583, 9545, 150587, 9562, 150707, 9563, 150743, 9565, 150767, 9567, 150797, 9582, 151213, 9583, 151243, 9585, 151247, 9592, 151303, 9593, 151343, 9595, 151397, 9597, 151507, 9702, 151517, 9703, 151597, 9705, 151703, 9707, 151717, 9712, 151783, 9713, 151787, 9715, 154213, 9717, 154243, 9742, 154247, 9745, 154267, 9747, 154303, 9762, 154313, 9763, 154387, 9765, 154543, 9782, 154747, 9783, 154787, 9785, 156217, 9792, 156307, 9793, 156347, 9795, 156593, 9797, 156703, 12020, 74567, 12021, 74587, 12024, 74597, 12026, 74707, 12028, 74713, 12029, 74717, 12030, 74747, 12031, 74797, 12034, 76207, 12036, 76213, 12038, 76243, 12039, 76283, 12050, 76303, 12051, 76343, 12054, 76367, 12056, 76387, 12058, 76507, 12059, 76543, 12070, 76597, 12074, 76717, 12076, 78203, 12078, 78283, 12079, 78307, 12120, 78317, 12121, 78347, 12124, 78367, 12126, 78517, 12128, 78583, 12129, 78593, 12130, 78707, 12131, 78713, 12134, 78787, 12136, 78797, 12138, 79283, 12139, 79367, 12150, 79393, 12151, 79397, 12154, 202021, 3629, 202031, 3630, 202121, 3634, 202129, 3636, 202471, 3638, 202621, 3639, 202639, 3650, 202679, 3651, 202859, 3654, 202879, 3656, 202921, 3658, 202931, 3670, 203039, 3674, 203051, 3676, 203429, 3678, 203431, 3679, 203459, 3820, 203659, 3824, 203821, 3826, 203921, 3828, 203971, 3829, 205031, 3830, 205129, 3831, 205151, 3834, 205171, 3836, 205421, 3838, 205651, 3839, 205879, 3850, 205951, 3854, 207029, 3856, 207079, 3858, 207121, 3859, 207139, 3870, 207479, 3871, 207629, 3874, 207671, 3876, 207679, 3878, 207821, 3879, 207931, 3920, 207971, 3921, 212029, 3924, 212039, 3926, 212131, 3928, 212479, 3930, 212651, 3934, 212671, 3936, 212851, 3938, 213029, 3939, 213079, 3950, 213131, 3951, 213139, 3954, 213659, 3956, 213821, 3958, 213859, 3959, 213929, 3970, 215051, 3971, 215179, 3974, 215459, 3976, 215471, 3978, 215659, 3979, 215851, 5020, 215921, 5024, 215939, 5026, 215959, 5028, 217121, 5029, 217421, 5030, 217429, 5031, 217439, 5034, 217829, 5036, 217859, 5038, 217979, 5050, 242059, 5054, 242129, 5056, 242171, 5058, 242479, 5070, 242621, 5071, 242629, 5074, 242639, 5076, 242659, 5078, 242971, 5079, 243031, 5120, 243121, 5121, 243421, 5124, 243431, 5126, 243479, 5128, 243631, 5129, 243671, 5130, 243829, 5131, 243839, 5134, 243851, 5136, 243871, 5138, 243931, 5139, 245029, 5150, 245039, 5151, 245071, 5154, 245129, 5156, 245131, 5158, 245171, 5159, 245471, 5170, 245621, 5174, 245629, 5176, 245639, 5178, 245671, 5420, 245821, 5421, 245851, 5424, 247031, 5426, 247421, 5428, 247439, 5429, 247451, 5430, 247651, 5434, 247829, 5436, 247879, 5438, 247939, 5439, 262051, 5450, 262079, 5451, 262121, 5454, 262139, 5456, 262151, 5458, 262459, 5459, 262621, 5470, 262651, 5474, 263071, 5476, 263129, 5478, 263171, 5620, 263429, 5621, 263621, 5624, 263651, 5626, 263821, 5628, 263951, 5629, 265021, 5630, 265079, 5631, 265129, 5634, 265151, 5636, 265451, 5638, 265459, 5650, 265471, 5654, 265621, 5656, 265831, 5658, 265871, 5670, 265921, 5671, 267131, 5674, 267139, 5676, 267431, 5678, 267439, 5679, 267451, 5820, 267479, 5824, 267629, 5826, 267671, 5828, 267679, 5829, 267829, 5830, 267929, 5831, 267959, 5834, 282059, 5836, 282071, 5838, 282439, 5850, 282671, 5854, 282679, 5856, 282851, 5858, 282959, 5859, 283051, 5870, 283079, 5871, 283121, 5874, 283139, 5876, 283159, 5878, 283631, 5920, 283639, 5921, 283831, 5924, 283859, 5926, 283979, 5928, 285031, 5929, 285071, 5930, 285079, 5931, 285121, 5934, 285139, 5936, 285151, 5938, 285179, 5950, 285421, 5951, 285451, 5954, 285629, 5956, 285631, 5958, 285839, 5959, 285871, 5970, 285979, 5971, 287059, 5974, 287159, 5976, 287179, 5978, 287629, 5979, 287671, 7020, 287821, 7021, 287851, 7024, 287921, 7026, 287939, 7028, 292021, 7029, 292079, 7030, 292459, 7031, 292471, 7034, 292631, 7036, 292679, 7038, 292879, 7050, 292921, 7051, 292979, 7054, 293021, 7056, 293071, 7058, 293129, 7059, 293179, 7070, 293431, 7071, 293459, 7074, 293621, 7076, 293639, 7078, 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Many thanks, Maximilian and Jean-Marc!

É.

[see here the same idea with odd/even terms]