Monday, January 28

Pseudo-binary ænlic primes

I discussed ænlic primes here. By pseudo-binary I mean decimal numbers that contain only digits zero and one. I will cover here three types of pseudo-binary ænlic primes:

1. Repunit primes. These are terms of A004022. They are ænlic primes by default — no other permutation of their digits is prime (because no other permutation of their digits exists). The repunit number (10^a-1)/9 is an ænlic prime for a = 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...

2. Primes composed of all ones except for one zero. The number (10^a-1)/9-10^b is an ænlic prime for (a,b) = (5,3), (6,4), (9,7), (26,13), (29,5), (33,13), (35,7), (39,23), (51,43), (53,37), (56,8), (74,44), (77,33), (80,11), (83,31), (89,39), (90,83), (92,53), (105,67), (107,11), (108,99), (110,61), (113,73), (126,44), (129,109), (134,80), (147,19), (149,15), (153,25), (164,15), (170,6), (174,62), (176,113), (177,167), (182,17), (185,37), (192,80), (194,26), (195,151), (201,53), (204,131), (215,119), (219,167), (221,181), (225,211), (227,223), (228,215), (230,32), (231,127), (234,208), (236,207), (246,76), (249,23), (251,147), (260,235), (263,151), (267,43), (270,128), (284,199), (285,53), (288,54), (297,13), (305,269), (314,278), (315,227), (317,125), (321,101), (324,320), (329,135), (330,289), (348,3), (363,83), (369,125), (371,131), (381,127), (387,379), (389,95), (396,262), (417,287), (419,119), (443,55), (446,342), (450,209), (453,421), (461,289), (467,459), (473,213), (477,53), (480,242), (491,31), (492,124), (497,423), (503,439), (504,486), (506,446), (512,83), (518,131), (521,419), (525,289), (539,483), (540,8), (545,247), (546,128), (548,284), (566,361), (567,103), (572,492), (579,83), (582,160), (588,396), (593,135), (605,79), (606,521), (609,163), (611,363), (618,208), (629,89), (632,293), (635,311), (641,331), (650,123), (653,33), (656,363), (657,175), (659,535), (674,626), (677,655), (681,509), (692,132), (698,209), (708,347), (710,25), (711,623), (713,451), (720,120), (729,697), (750,146), (752,98), (765,167), (770,530), (779,355), (780,84), (791,623), (803,459), (804,148), (806,13), (809,105), (816,336), (828,816), (833,491), (836,399), (837,425), (845,35), (848,521), (849,691), (857,545), (858,627), (861,293), (870,112), (872,475), (875,771), (885,703), (890,87), (893,85), (897,487), (912,35), (914,168), (923,71), (935,415), (939,295), (945,169), (954,897), (956,746), (969,311), (984,695), (993,397), (1001,259), (1017,151), (1019,735), (1020,843), (1022,629), (1029,923), (1037,289), (1040,11), (1052,559), (1067,863), (1070,750), (1077,695), (1083,631), (1085,463), (1086,1073), (1091,511), (1095,655), (1097,565), (1101,997), (1110,626), (1112,889), (1124,561), (1127,711), (1136,737), (1139,999), (1140,310), (1146,526), (1148,119), (1149,253), (1155,455), (1160,746), (1161,61), (1163,227), (1173,187), (1175,983), (1184,347), (1196,557), (1199,623), (1202,194), (1203,583), (1208,189), (1217,35), (1230,1190), (1232,277), (1235,959), (1254,1035), (1256,1083), (1259,1143), (1265,735), (1274,1199), (1277,1113), (1287,607), (1289,755), (1302,446), (1307,1235), (1320,239), (1328,853), (1331,139), (1338,905), (1346,1237), (1353,907), (1361,1329), (1365,635), (1373,1269), (1376,608), (1377,83), (1382,547), (1385,545), (1392,638), (1407,383), (1409,245), (1421,377), (1433,1013), (1451,1239), (1466,999), (1470,1159), (1472,1344), (1473,271), (1481,1059), (1485,325), (1487,527), (1509,467), (1511,1231), (1514,1039), (1515,443), (1517,1239), (1521,229), (1523,947), (1526,529), (1529,421), (1532,1205), (1533,749), (1541,1123), (1548,1388), (1550,828), (1553,1351), (1554,684), (1556,746), (1566,738), (1568,1001), (1572,1244), (1586,843), (1589,1017), (1598,168), (1604,289), (1613,319), (1617,139), (1620,4), (1625,69), (1635,1595), (1641,841), (1643,427), (1658,1299), (1662,1486), (1665,479), (1670,1214), (1686,191), (1688,114), (1689,637), (1700,524), (1706,3), (1707,383), (1709,1607), (1716,83), (1721,369), (1725,223), (1731,119), (1737,941), (1739,1435), (1757,1491), (1763,1019), (1766,893), (1770,104), (1773,661), (1793,601), (1796,1003), (1797,421), (1802,380), (1805,229), (1809,13), (1811,23), (1815,719), (1818,1181), (1821,751), (1836,177), (1850,143), (1862,1407), (1880,294), (1904,1195), (1905,1891), (1907,1839), (1914,1012), (1926,1226), (1929,473), (1934,1017), (1955,303), (1956,416), (1958,1081), (1970,1755), (1973,331), (1985,1355), (1986,1426), (1992,1083), (1995,535), (2010,1518), (2016,396), (2019,683), (2021,1977), (2034,1186), (2036,476), (2037,1451), (2039,1679), (2040,1270), (2048,543), (2058,2022), (2060,752), (2063,1071), (2067,1895), (2069,2045), (2070,2001), (2084,1065), (2085,1865), (2091,491), (2094,25), (2097,1463), (2105,1011), (2112,41), (2115,1007), (2123,531), (2124,1345), (2127,2063), (2135,79), (2136,1214), (2138,62), (2141,57), (2147,1515), (2166,1426), (2178,1320), (2192,120), (2193,509), (2195,1723), (2198,1739), (2204,895), (2205,611), (2208,1648), (2210,1697), (2213,1523), (2217,593), (2219,1643), (2225,481), (2234,755), (2235,991), (2249,2123), (2255,1711), (2259,467), (2261,1909), (2262,13), (2264,2113), (2265,1351), (2289,823), (2306,1038), (2309,1305), (2316,594), (2321,553), (2324,1089), (2325,907), (2328,1015), (2336,1403), (2339,1703), (2345,665), (2354,901), (2366,1886), (2373,461), (2379,107), (2385,1225), (2387,159), (2388,1713), (2390,1353), (2394,152), (2397,1295), (2400,463), (2403,203), (2409,5), (2411,459), (2417,257), (2420,1700), (2426,1389), (2430,2106), (2432,857), (2433,751), (2439,2239), (2441,599), (2447,943), (2451,1543), (2457,1609), (2465,1339), (2472,1648), (2478,232), (2481,1781), (2486,1817), (2489,969), (2490,1304), (2498,1541), (2501,2117), (2508,864), (2513,673), (2519,1679), (2520,223), (2522,2466), (2526,1522), (2528,2067), (2535,1567), (2537,1889), (2546,108), (2549,757), (2550,1175), (2552,889), (2553,1933), (2561,863), (2564,2363), (2565,359), (2567,2039), (2571,839), (2574,1322), (2577,1129), (2585,967), (2594,313), (2595,607), (2597,2019), (2598,974), (2601,2329), (2606,1229), (2610,1192), (2612,1987), (2616,1281), (2618,1827), (2621,2607), (2628,1762), (2633,2073), (2636,903), (2646,1011), (2661,1321), (2670,810), (2673,2095), (2688,2672), (2693,1033), (2694,1187), (2696,2568), (2708,855), (2733,1537), (2759,1119), (2766,2361), (2769,383), (2774,558), (2784,1545), (2789,1019), (2793,2023), (2795,1863), (2796,2563), (2802,2724), (2804,1610), (2807,1455), (2816,449), (2822,356), (2826,1362), (2828,635), (2844,2126), (2846,2597), (2856,2573), (2858,593), (2861,1197), (2865,991), (2867,991), (2873,1801), (2877,1085), (2882,1454), (2883,779), (2885,1065), (2889,1579), (2897,2701), (2903,1031), (2904,894), (2912,165), (2925,197), (2927,447), (2931,1063), (2939,2755), (2948,727), (2954,2369), (2963,2711), (2966,1316), (2981,421), (2987,1155), (2993,555), (2996,608), (2997,2777), (3008,1807), (3009,2405), (3020,2891), (3027,823), (3033,361), (3041,1847), (3044,2694), (3047,1359), (3066,2488), (3075,3031), (3080,1537), (3081,817), (3083,2107), (3111,2519), (3114,12), (3116,848), (3126,1936), (3128,1927), (3129,343), (3131,2847), (3135,1663), (3137,2399), (3156,2089), (3165,2035), (3170,896), (3173,875), (3186,2354), (3191,1047), (3201,1219), (3209,2407), (3213,2951), (3233,2885), (3239,983), (3242,1188), (3246,1819), (3249,3173), (3255,3223), (3260,2767), (3263,1535), (3266,798), (3267,1699), (3282,1098), (3290,1988), (3296,2093), (3306,1708), (3311,3023), (3312,2481), (3315,919), (3320,1875), (3324,1342), (3332,2474), (3341,2493), (3348,1870), (3353,2917), (3356,739), (3357,871), (3365,1673), (3369,3059), (3381,2693), (3398,2421), (3399,2959), (3401,2247), (3410,193), (3422,427), (3426,2937), (3452,2364), (3459,1255), (3467,2375), (3477,3253), (3485,783), (3489,73), (3494,1003), (3506,2407), (3509,123), (3524,539), (3531,487), (3533,985), (3536,1847), (3537,847), (3539,1643), (3545,1243), (3561,593), (3564,3341), (3570,3482), (3578,149), (3587,1791), (3593,427), (3597,937), (3609,1105), (3614,887), (3615,1535), (3618,3464), (3629,3285), (3641,2459), (3648,2379), (3657,2615), (3666,3056), (3669,1045), (3677,3015), (3686,3103), (3689,3107), (3690,2692), (3698,401), (3699,2047), (3702,2292), (3704,2503), (3708,2718), (3714,1189), (3723,1343), (3725,783), (3734,3669), (3737,293), (3747,263), (3755,827), (3759,671), (3761,1019), (3777,2821), (3779,3011), (3782,2231), (3783,3335), (3785,821), (3791,799), (3794,63), (3795,2231), (3798,2092), (3812,1435), (3821,1769), (3825,1187), (3834,2088), (3837,1235), (3851,3271), (3855,607), (3858,66), (3864,3224), (3866,307), (3867,1255), (3872,1982), (3876,3329), (3881,2461), (3887,3791), (3888,1640), (3891,1799), (3896,3377), (3920,848), (3923,779), (3924,1629), (3927,1879), (3936,2181), (3939,1159), (3941,1811), (3942,3348), (3947,2307), (3968,2964), (3969,653), (3975,3223), (3980,926), (3989,3385), (3996,3846), (4004,3722), (4005,295), (4017,743), (4020,2780), (4046,2039), (4053,3251), (4058,3371), (4059,3151), (4061,2773), (4062,1576), (4064,1435), (4067,3243), (4068,590), (4073,2597), (4091,3883), (4097,567), (4100,2737), (4101,2483), (4106,947), (4113,2231), (4115,643), (4118,377), (4139,3319), (4142,2821), (4148,889), (4149,1909), (4154,1620), (4160,3499), (4169,1309), (4179,1063), (4184,3667), (4188,452), (4197,3167), (4200,1198), (4203,427), (4211,1939), (4214,1227), (4218,3115), (4221,503), (4226,318), (4229,1427), (4232,2198), (4235,695), (4238,3444), (4241,977), (4245,835), (4247,3679), (4256,551), (4257,2287), (4259,3195), (4262,1970), (4263,1775), (4265,1489), (4266,249), (4269,2543), (4275,779), (4277,4091), (4281,3961), (4289,2387), (4293,4085), (4298,1166), (4301,1021), (4304,4116), (4307,3347), (4316,2149), (4325,461), (4332,1277), (4334,1935), (4337,2585), (4353,911), (4355,2039), (4356,2452), (4358,753), (4359,2407), (4361,1049), (4365,433), (4368,3505), (4370,2113), (4376,863), (4377,2311), (4395,4339), (4400,987), (4401,2993), (4403,3651), (4409,1781), (4418,2735), (4424,276), (4433,4319), (4436,4003), (4437,377), (4449,3119), (4451,399), (4454,240), (4455,2527), (4460,2165), (4463,1167), (4478,2304), (4481,3209), (4484,3163), (4485,149), (4490,2162), (4493,3767), (4499,319), (4505,427), (4506,1404), (4517,587), (4521,4141), (4526,3689), (4535,351), (4538,2917), (4542,3615), (4547,1743), (4548,2601), (4553,439), (4556,3509), (4559,1839), (4560,1397), (4565,4149), (4568,651), (4569,1061), (4571,2687), (4578,1734), (4580,846), (4581,541), (4586,1976), (4589,2197), (4593,503), (4608,2017), (4617,203), (4619,1455), (4629,3797), (4638,1778), (4641,4327), (4643,2135), (4646,1638), (4649,3149), (4656,244), (4658,1813), (4686,1206), (4700,1007), (4703,2559), (4707,1399), (4709,1753), (4713,3883), (4725,3079), (4728,1648), (4730,3371), (4733,3719), (4736,339), (4739,3115), (4740,1004), (4745,555), (4746,2662), (4748,824), (4752,240), (4757,35), (4761,3671), (4769,2495), (4773,2183), (4776,3427), (4784,270), (4787,4495), (4793,217), (4796,239), (4802,3989), (4805,715), (4818,2755), (4820,1099), (4821,2351), (4826,1653), (4829,4721), (4833,3359), (4835,879), (4851,511), (4853,1319), (4854,4423), (4856,4477), (4878,1071), (4881,1373), (4883,739), (4884,748), (4886,327), (4898,3849), (4905,725), (4908,232), (4917,1399), (4919,3951), (4925,2659), (4929,4297), (4931,311), (4947,547), (4952,4455), (4955,831), (4958,1553), (4961,153), (4964,4205), (4977,3875), (4989,359), (5001,3403), (5006,2591), (5009,2893), (5018,3570), (5019,391), (5025,1025), (5027,783), (5030,3686), (5037,1261), (5045,3219), (5061,3523), (5063,527), (5064,1266), (5066,4099), (5073,989), (5076,402), (5084,3252), (5093,1239), (5096,2039), (5097,4675), (5099,1055), (5102,2082), (5108,4176), (5114,335), (5121,3103), (5123,2935), (5126,2802), (5135,1999), (5139,2803), (5148,4711), (5153,2893), (5159,4799), (5165,3333), (5168,4911), (5180,4565), (5186,3548), (5187,851), (5189,4755), (5193,4717), (5195,3739), (5201,2637), (5211,1687), (5219,1391), (5226,3709), (5229,3577), (5232,4567), (5235,1007), (5237,5051), (5243,3743), (5249,1401), (5250,179), (5252,1485), (5259,1775), (5262,267), (5265,4423), (5273,5255), (5276,3979), (5285,655), (5294,2927), (5298,4206), (5300,559), (5316,5302), (5331,1939), (5333,3187), (5337,5323), (5339,2411), (5345,3893), (5348,3908), (5351,4159), (5363,2551), (5373,205), (5381,2487), (5382,4234), (5384,4317), (5388,4150), (5399,3143), (5406,1736), (5408,4602), (5409,293), (5411,1979), (5421,5263), (5433,5203), (5438,2693), (5441,5171), (5445,4349), (5453,3793), (5457,5375), (5466,1436), (5468,167), (5474,4931), (5477,5285), (5481,3853), (5483,2603), (5487,367), (5489,4013), (5493,2501), (5496,3928), (5498,1819), (5505,3371), (5520,2747), (5526,3918), (5529,2483), (5531,3587), (5535,2911), (5544,4507), (5553,5411), (5561,563), (5565,2341), (5573,1501), (5576,2952), (5589,5047), (5619,623), (5622,2668), (5625,13), (5639,4959), (5651,2123), (5658,1435), (5660,5359), (5682,4722), (5694,758), (5697,1241), (5717,3561), (5720,3433), (5721,4327), (5726,3758), (5733,3277), (5741,717), (5747,263), (5750,1710), (5754,2301), (5756,5323), (5763,203), (5765,2733), (5769,4387), (5774,488), (5777,4469), (5781,5183), (5786,5459), (5795,4567), (5807,5567), (5816,5293), (5822,4983), (5826,2469), (5829,3895), (5837,2343), (5841,5839), (5843,31), (5847,2143), (5874,3064), (5892,5790), (5921,3513), (5925,2309), (5927,4679), (5928,5111), (5934,2729), (5936,3221), (5937,3047), (5943,5255), (5945,3645), (5948,5712), (5955,2423), (5957,5593), (5958,2628), (5963,795), (5964,1648), (5966,3183), (5969,4705), (5975,439), (5978,5111), (5981,3719), (5982,2593), (5987,947), (5991,3119), (5993,5625), (5996,728), (6000,5672), (6006,1988), (6017,821), (6018,421), (6023,4911), (6026,669), (6027,1567), (6032,5094), (6039,2879), (6041,251), (6048,1605), (6056,3167), (6066,2939), (6083,1155), (6092,5461), (6096,548), (6098,1831), (6101,3433), (6104,485), (6107,779), (6119,2159), (6126,1474), (6129,2177), (6131,1543), (6132,961), (6140,2279), (6141,3521), (6155,3267), (6164,884), (6173,4181), (6177,2921), (6186,2566), (6188,773), (6189,6119), (6194,409), (6200,3455), (6225,181), (6227,683), (6230,5977), (6234,338), (6236,1063), (6237,3955), (6239,1279), (6240,4833), (6242,1622), (6249,3115), (6266,2123), (6269,253), (6281,173), (6284,3582), (6299,5987), (6320,3469), (6324,2989), (6330,1691), (6350,2015), (6354,1427), (6359,1295), (6366,671), (6371,5127), (6372,3430), (6374,3857), (6375,2567), (6377,2893), (6383,319), (6389,3317), (6393,4663), (6395,5707), (6396,5848), (6402,4246), (6405,1427), (6407,6287), (6414,1080), (6416,3267), (6417,5273), (6420,5311), (6423,3823), (6429,6191), (6432,5852), (6434,5553), (6435,3239), (6437,4253), (6449,2083), (6452,1760), (6455,3055), (6459,1891), (6473,157), (6482,1452), (6486,2018), (6489,5305), (6500,4487), (6501,3589), (6506,2789), (6507,2171), (6513,5891), (6522,6289), (6525,6431), (6536,3417), (6546,3334), (6548,3091), (6557,1385), (6558,6404), (6570,5538), (6572,2161), (6581,6199), (6584,1013), (6585,5383), (6587,107), (6588,240), (6596,1089), (6605,1653), (6627,827), (6633,1747), (6635,415), (6651,1087), (6659,1743), (6660,105), (6675,607), (6689,6613), (6690,338), (6701,4249), (6707,2887), (6710,624), (6711,6143), (6714,3505), (6740,2568), (6741,2047), (6743,6503), (6749,6339), (6750,452), (6758,4621), (6759,4319), (6761,2979), (6762,881), (6764,2906), (6767,3823), (6785,5979), (6788,2600), (6821,4921), (6822,5648), (6825,2347), (6827,3583), (6828,1702), (6830,4495), (6831,3791), (6833,1791), (6836,5581), (6837,4441), (6846,1433), (6849,2935), (6852,746), (6860,4998), (6866,6699), (6867,5755), (6873,1867), (6876,549), (6887,6103), (6888,1856), (6893,1607), (6896,6092), (6932,6859), (6935,6183), (6936,6933), (6938,2425), (6950,2226), (6956,2679), (6974,4373), (6977,6947), (6978,3676), (6981,1789), (6989,2783), (6990,5726), (6999,2191), (7001,3729), (7005,4973), (7008,1268), (7010,4650), (7013,5461), (7014,1874), (7022,3512), (7028,3236), (7029,6775), (7043,531), (7049,1829), (7050,6471), (7053,2135), (7061,6047), (7068,145), (7077,5869), (7079,4879), (7080,4350), (7082,6325), (7085,5737), (7088,4958), (7092,2147), (7094,3489), (7098,1036), (7112,4171), (7115,3739), (7119,1391), (7136,1957), (7139,3823), (7145,2567), (7146,6039), (7152,6245), (7163,1047), (7169,1103), (7170,5173), (7173,6703), (7178,2785), (7179,1723), (7181,4003), (7185,3089), (7190,4442), (7193,4513), (7202,6959), (7206,2802), (7212,4390), (7217,2473), (7218,472), (7220,2180), (7224,2888), (7226,6823), (7227,6439), (7229,6423), (7232,143), (7235,1851), (7236,2682), (7241,1817), (7260,862), (7262,6869), (7272,1805), (7290,4459), (7296,5549), (7304,3072), (7307,3403), (7316,6569), (7323,2407), (7325,3513), (7326,6493), (7329,2485), (7343,3359), (7344,825), (7346,1169), (7347,1135), (7349,4937), (7355,6907), (7359,6335), (7361,3143), (7364,2372), (7367,6047), (7368,872), (7374,1446), (7376,7242), (7379,2591), (7380,4138), (7389,227), (7400,135), (7409,2345), (7412,362), (7415,3775), (7428,302), (7434,2383), (7439,2991), (7442,1333), (7445,4595), (7451,6343), (7452,7345), (7454,561), (7458,527), (7467,4543), (7484,5043), (7487,6143), (7488,3785), (7490,1129), (7491,4751), (7497,4429), (7502,6000), (7506,5374), (7508,3019), (7509,3445), (7521,3781), (7529,1521), (7550,1046), (7574,6115), (7577,2725), (7587,6895), (7589,971), (7593,7421), (7596,6414), (7598,4201), (7610,4339), (7613,4217), (7620,3856), (7629,1261), (7640,7026), (7641,4549), (7643,1383), (7653,245), (7656,3296), (7658,5670), (7667,5515), (7674,5379), (7676,2113), (7677,5791), (7683,7403), (7686,3268), (7689,2101), (7700,2772), (7701,7237), (7707,2503), (7710,4647), (7713,2947), (7719,1879), (7728,2847), (7730,1644), (7731,5543), (7736,872), (7739,2203), (7742,4370), (7745,3975), (7757,3277), (7760,329), (7761,793), (7775,3455), (7784,4883), (7785,4667), (7787,1431), (7797,6445), (7803,4243), (7806,2363), (7808,5462), (7809,1871), (7817,4945), (7824,2168), (7826,6422), (7841,1607), (7845,2599), (7847,679), (7850,984), (7853,2695), (7856,686), (7857,875), (7869,4271), (7872,3167), (7875,5315), (7880,6029), (7883,2903), (7896,3174), (7904,500), (7905,6427), (7907,1027), (7914,2253), (7925,3861), (7929,749), (7937,1637), (7967,6463), (7977,1313), (7982,7069), (8000,938), (8001,2533), (8006,2522), (8009,2515), (8013,593), (8033,1977), (8036,1388), (8042,174), (8046,5769), (8051,6439), (8057,1673), (8064,69), (8066,4572), (8072,7785), (8073,3097), (8082,1410), (8087,687), (8093,7397), (8096,2307), (8097,5149), (8102,6425), (8105,1099), (8114,6689), (8117,735), (8120,2791), (8121,6131), (8123,7295), (8130,558), (8133,4691), (8139,7915), (8141,357), (8144,1459), (8153,4011), (8156,4421), (8162,2570), (8166,2318), (8184,4743), (8186,6811), (8187,3079), (8193,175), (8211,7963), (8213,2681), (8220,6964), (8231,8007), (8235,3583), (8237,5821), (8241,4571), (8246,2501), (8250,2015), (8256,5007), (8259,755), (8270,2669), (8276,5069), (8283,6419), (8297,4893), (8301,997), (8307,2795), (8309,2499), (8312,2219), (8315,6991), (8327,5335), (8339,6243), (8342,902), (8348,613), (8352,1263), (8354,2099), (8360,4410), (8361,5351), (8370,2376), (8372,817), (8375,7919), (8384,5201), (8385,4105), (8402,1963), (8420,8052), (8421,6067), (8441,7277), (8444,6443), (8448,5731), (8457,1811), (8460,6940), (8462,4016), (8466,5958), (8469,37), (8477,8003), (8480,1915), (8499,1555), (8504,122), (8508,2812), (8513,1137), (8525,5037), (8526,442), (8537,669), (8541,5857), (8543,7839), (8544,1658), (8549,2021), (8558,7677), (8561,6347), (8562,7486), (8564,507), (8573,5693), (8574,6126), (8577,2089), (8579,751), (8586,8246), (8592,1383), (8595,4255), (8600,4293), (8601,5497), (8604,7233), (8612,6738), (8613,1381), (8619,6223), (8622,7181), (8625,985), (8627,7883), (8631,2399), (8634,4076), (8636,3926), (8652,8280), (8657,2415), (8660,4609), (8661,4261), (8667,8507), (8669,1161), (8673,2305), (8685,1187), (8688,6464), (8691,7751), (8693,7875), (8697,3769), (8706,6124), (8717,4015), (8721,2119), (8727,7943), (8738,5516), (8741,5289), (8744,6537), (8753,383), (8756,6413), (8757,4397), (8763,623), (8768,6371), (8771,8471), (8774,5369), (8786,4122), (8805,2383), (8813,6757), (8825,4421), (8828,2642), (8835,7867), (8837,1407), (8840,2666), (8841,2233), (8843,1595), (8846,5768), (8847,6895), (8858,563), (8864,1697), (8865,1495), (8868,6236), (8873,5557), (8880,7871), (8885,1543), (8891,5819), (8892,8199), (8894,8852), (8897,6195), (8907,6727), (8909,5857), (8915,5223), (8933,701), (8937,5417), (8939,4427), (8945,8575), (8955,883), (8961,4153), (8972,8780), (8978,1467), (8993,591), (9003,8819), (9005,8553), (9011,4599), (9014,1250), (9015,7055), (9018,7367), (9020,3867), (9024,5566), (9029,7821), (9041,8657), (9044,480), (9045,4205), (9053,175), (9054,7918), (9059,1923), (9083,2007), (9089,4225), (9090,1328), (9093,2791), (9095,7263), (9099,115), (9101,8231), (9102,622), (9107,4219), (9110,2605), (9111,6023), (9114,1784), (9120,4465), (9126,9111), (9128,3626), (9132,844), (9140,3798), (9143,4759), (9144,53), (9146,6919), (9147,8935), (9150,3919), (9168,971), (9173,4219), (9176,2947), (9182,3721), (9185,6435), (9186,7994), (9189,8377), (9203,7587), (9213,6311), (9219,6791), (9225,233), (9230,878), (9237,157), (9246,9129), (9248,8531), (9254,6786), (9264,8126), (9270,812), (9275,9123), (9279,895), (9281,1753), (9285,9067), (9288,2491), (9291,2323), (9293,6335), (9294,9126), (9300,724), (9311,5951), (9317,3867), (9320,6907), (9321,4283), (9353,7267), (9360,1702), (9362,3594), (9366,5698), (9368,2423), (9369,1225), (9378,5258), (9380,4370), (9387,4403), (9393,1985), (9401,857), (9402,6521), (9405,8407), (9408,8010), (9411,2471), (9413,879), (9417,7159), (9419,675), (9428,7158), (9429,9017), (9434,1154), (9435,2171), (9443,6471), (9464,4601), (9474,7370), (9477,2345), (9480,6171), (9491,1347), (9500,5303), (9506,2372), (9510,4548), (9512,4022), (9522,9489), (9525,1843), (9534,2670), (9537,7513), (9543,2863), (9545,6737), (9551,703), (9558,3878), (9563,6039), (9570,7703), (9573,1955), (9575,4335), (9578,4385), (9579,1403), (9584,1033), (9585,4285), (9591,6239), (9600,5725), (9603,7111), (9618,838), (9620,8705), (9623,6799), (9627,2843), (9632,2546), (9636,689), (9641,4453), (9645,2095), (9653,6621), (9656,7777), (9660,8459), (9665,6761), (9666,1828), (9668,8046), (9675,3127), (9684,5344), (9687,3583), (9689,5539), (9698,8604), (9701,791), (9704,1526), (9710,5847), (9714,1707), (9716,1533), (9726,2468), (9732,910), (9741,3821), (9753,5515), (9755,8455), (9758,1159), (9761,7113), (9767,5103), (9771,67), (9773,1181), (9774,7092), (9794,6633), (9797,3635), (9806,5637), (9809,6909), (9813,1453), (9816,619), (9818,2816), (9819,6695), (9827,6751), (9830,6111), (9834,2639), (9843,8219), (9849,2933), (9851,6567), (9861,2053), (9863,855), (9864,3981), (9867,5239), (9878,7799), (9894,9519), (9896,5103), (9899,3219), (9902,8472), (9905,3785), (9915,895), (9917,7429), (9918,9568), (9920,3863), (9933,5663), (9950,3469), (9966,834), (9981,9209), (9998,5682), (10002,2872), ...

3. Primes composed of all ones except for two zeros. The number (10^a-1)/9-10^b-10^c is an ænlic prime for (a,b,c) = (12,9,1), (24,19,11), (384,353,225), ... If there are more ænlic primes of this form, then a > 2050.

I'm not aware of any ænlic primes composed of all ones except for three (or more) zeros.

Tuesday, January 22

Long time comin'


I first queried The Mudcat Café about this piece of music in August 1999. The fact that I had been unable since then to elicit a credible lead [see the final paragraph in my blog here, the Google+ posts here and (a reprise of the blog item) here] has been a thorn in my side! This evening I was going through some random old-time favourites in my music library and I thought again to look for it. Google showed me the Mudcat link and I went there to see how long ago I had posted my question. As I advanced the thread I reached a January 2017 guest posting with the above YouTube link.

To be fair, Anderson's voice wasn't exactly how my mind had recreated the memory of it, but there was little doubt that this was the song that I had last heard on the radio circa 1970 — almost fifty years ago. The YouTube audio had been up since September 2013. Three-plus years for someone to link it to the Mudcat thread and two more years for me to notice it.

Over the years I had convinced myself that I would never know the artist/song before I died. But my faith in the internet has been somewhat restored and I can cross this difficult things-to-find task off my bucket list.

Thursday, January 17

My 200th Leyland prime find


In August 2016 I mentioned my 100th Leyland prime find. This morning I found my 200th. The graph (click on it for a better view) shows the number of decimal digits of those 200 primes as a function of the discovery date (the calendar years mark the start of those years). One may note the periods of relative inactivity where I was burdening my processors with other tasks. The upper-right plateau constitutes my "long compute" which will come in at just under two years when it's done in April of this year.