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Old 2021-02-22, 13:40   #122
sweety439
 
Nov 2016

22·3·5·47 Posts
Default

Code:
Minimal set of emirps in decimal:

{13, 17, 31, 37, 71, 73, 79, 97, 149, 199, 359, 389, 941, 953, 983, 991, 1009, 1021, 1061, 1069, 1091, 1109, 1151, 1181, 1201, 1229, 1259, 1511, 1559, 1601, 1619, 1669, 1811, 1901, 3023, 3049, 3083, 3203, 3299, 3343, 3433, 3463, 3469, 3583, 3643, 3803, 3853, 3929, 7027, 7057, 7207, 7457, 7507, 7547, 7577, 7687, 7757, 7867, 9001, 9011, 9029, 9161, 9209, 9221, 9293, 9349, 9403, 9439, 9521, 9551, 9601, 9643, 9661, 9923, 10159, 10859, 10889, 11159, 11161, 11621, 12119, 12241, 12611, 12641, 12689, 12809, 12841, 14081, 14221, 14251, 14551, 14621, 14821, 15241, 15289, 15461, 15541, 15661, 16111, 16451, 16481, 16651, 16829, 18041, 18089, 18169, 18269, 18461, 18691, 18859, 19681, 30029, 32009, 32233, 32353, 32369, 32563, 32633, 32693, 32933, 33029, 33223, 33329, 33623, 33863, 33923, 35323, 35363, 36209, 36269, 36353, 36523, 36833, 39623, 70667, 74077, 76607, 77047, 90023, 90059, 90089, 90263, 90499, 90821, 90989, 91121, 91129, 92003, 92033, 92119, 92189, 92333, 92369, 92459, 92489, 92639, 92861, 92899, 92959, 93629, 94559, 94889, 95009, 95101, 95111, 95429, 95549, 95801, 95881, 95929, 96181, 96263, 96281, 96289, 96323, 96329, 98009, 98081, 98129, 98251, 98269, 98299, 98429, 98621, 98801, 98849, 98909, 98999, 99289, 99409, 99829, 99989, 100411, 101119, 101141, 104801, 108401, 108881, 111119, 111211, 111829, 111869, 112111, 114001, 114041, 116911, 119611, 121421, 121921, 124121, 125551, 125651, 125821, 126551, 126851, 128521, 128551, 129121, 129281, 140411, 141101, 144481, 144541, 145441, 145861, 146161, 146581, 152981, 155521, 155581, 155621, 155821, 155851, 156521, 158551, 158621, 158981, 161641, 162881, 168541, 182921, 184441, 185291, 185551, 185641, 185681, 185819, 186581, 186889, 188261, 188609, 188801, 189251, 189851, 192581, 302609, 302629, 305603, 305633, 306329, 306503, 322229, 322249, 322649, 323333, 324293, 328883, 329663, 330053, 333253, 333323, 333563, 335633, 335653, 336503, 336533, 338383, 340453, 340909, 340999, 344053, 344293, 344453, 344843, 348443, 349399, 349493, 350033, 350443, 350663, 352333, 354043, 354443, 356533, 362293, 362339, 362449, 363683, 364909, 365333, 366053, 366239, 366293, 366409, 366923, 383683, 383833, 386363, 386383, 388823, 392263, 392423, 392443, 392663, 394943, 394993, 399493, 700067, 704447, 704747, 707767, 724747, 725587, 725827, 727487, 727877, 728527, 728747, 740087, 744407, 746267, 746747, 747407, 747427, 747647, 747827, 748877, 755267, 760007, 762557, 762647, 766477, 767707, 774667, 777787, 777877, 778727, 778777, 778847, 780047, 780887, 784727, 785527, 787777, 788087, 904459, 904663, 906203, 906881, 906949, 908549, 908959, 909043, 909463, 909599, 911101, 911111, 915869, 918581, 918829, 921629, 922223, 922499, 923603, 924299, 926129, 926203, 928111, 928159, 928289, 928469, 928819, 928859, 932663, 933263, 940469, 942223, 942569, 944263, 945089, 945809, 946223, 946459, 946949, 948659, 949609, 949649, 950569, 951089, 951829, 951859, 952859, 954409, 954649, 954869, 956569, 956689, 956849, 958159, 958259, 958829, 959489, 959689, 959809, 959869, 960889, 964049, 964829, 965059, 965189, 965249, 965659, 965969, 968111, 968459, 968519, 968959, 969569, 980159, 980549, 981569, 982829, 984959, 986659, 986959, 988069, 988681, 992429, 993943, 994229, 995699, 995909, 996599, 999043, 1005581, 1040141, 1041041, 1054441, 1055881, 1080851, 1100441, 1111921, 1119121, 1124141, 1126661, 1144441, 1164461, 1164641, 1165889, 1166441, 1185689, 1219111, 1222681, 1226581, 1228651, 1248881, 1252681, 1255861, 1262581, 1268261, 1286881, 1291111, 1298581, 1298651, 1400051, 1401401, 1410401, 1412461, 1414211, 1414261, 1424681, 1440011, 1444411, 1444501, 1446611, 1464461, 1464611, 1466461, 1486561, 1488481, 1500041, 1508851, 1522681, 1568221, 1568921, 1568951, 1580801, 1585261, 1588051, 1588681, 1589561, 1595861, 1598651, 1624141, 1624661, 1625851, 1628621, 1642141, 1644611, 1644641, 1646641, 1656841, 1659851, 1659881, 1662281, 1664261, 1666211, 1685521, 1685951, 1808581, 1822661, 1848841, 1850129, 1852621, 1855001, 1856221, 1858081, 1858921, 1862221, 1862251, 1862521, 1864241, 1868851, 1882891, 1885501, 1886821, 1888189, 1888421, 1888981, 1889561, 1898881, 1982881, 3009443, 3033053, 3033533, 3065533, 3066229, 3099443, 3220669, 3232429, 3232963, 3233663, 3242429, 3249443, 3282283, 3284483, 3305063, 3322069, 3326629, 3326663, 3332363, 3334099, 3334999, 3339949, 3349999, 3353303, 3355603, 3365563, 3369409, 3422429, 3422453, 3424249, 3424283, 3434099, 3440399, 3442409, 3449003, 3449423, 3449903, 3449909, 3452453, 3494009, 3499099, 3499499, 3499999, 3503303, 3522263, 3542243, 3542543, 3600563, 3605033, 3622253, 3626683, 3628663, 3632333, 3632663, 3632963, 3634949, 3634999, 3649999, 3650063, 3655633, 3662363, 3663323, 3663949, 3664249, 3664499, 3666233, 3668263, 3669499, 3692323, 3692363, 3699499, 3822823, 3824243, 3828683, 3838883, 3844823, 3866263, 3868283, 3888383, 3940099, 3944909, 3969649, 7004087, 7004477, 7004677, 7008047, 7008887, 7047787, 7048477, 7067077, 7074677, 7080077, 7224647, 7224667, 7227287, 7228477, 7242727, 7246727, 7256657, 7262467, 7262677, 7264277, 7272427, 7272647, 7272887, 7276427, 7278827, 7282487, 7287277, 7288727, 7400467, 7408007, 7422887, 7426477, 7426667, 7427447, 7427477, 7427887, 7428887, 7444277, 7446277, 7447067, 7447247, 7447267, 7447777, 7462277, 7462727, 7464227, 7474877, 7477667, 7482287, 7522267, 7526557, 7556257, 7562567, 7565567, 7566527, 7607447, 7622257, 7622767, 7627447, 7640047, 7642627, 7647667, 7652657, 7655657, 7664227, 7666247, 7667467, 7667747, 7672267, 7700807, 7707607, 7722647, 7724447, 7724627, 7726447, 7727827, 7744007, 7746247, 7747247, 7748227, 7748407, 7762627, 7764007, 7764707, 7777447, 7784747, 7788787, 7804007, 7822847, 7827227, 7842827, 7848787, 7877407, 7878487, 7878877, 7882247, 7882727, 7887247, 7888007, 7888247, 9000049, 9004469, 9004669, 9004943, 9015689, 9042443, 9046589, 9048509, 9049633, 9058409, 9059969, 9065449, 9065669, 9068069, 9081659, 9084469, 9088589, 9088699, 9090649, 9094469, 9094493, 9094649, 9094669, 9099443, 9099469, 9118589, 9185689, 9210581, 9222569, 9225589, 9225989, 9226549, 9226603, 9226859, 9229699, 9229949, 9242243, 9242323, 9242423, 9244969, 9252589, 9252949, 9256589, 9256969, 9264449, 9266233, 9268559, 9269699, 9284449, 9285569, 9292949, 9298889, 9299669, 9400009, 9404849, 9406549, 9408569, 9424243, 9424663, 9429499, 9429869, 9444629, 9444829, 9445609, 9446659, 9446989, 9452269, 9456049, 9456229, 9460849, 9460909, 9464099, 9464849, 9464909, 9469693, 9480649, 9484049, 9484649, 9492529, 9492929, 9493663, 9494363, 9494689, 9496259, 9498959, 9499229, 9499333, 9506459, 9506969, 9508669, 9509959, 9522269, 9525589, 9526949, 9545099, 9545969, 9546059, 9546599, 9550669, 9552299, 9555569, 9555599, 9555859, 9556699, 9556889, 9556969, 9558629, 9559699, 9560669, 9561809, 9562499, 9566449, 9568859, 9569089, 9581189, 9584089, 9584699, 9585559, 9585889, 9586229, 9588659, 9588869, 9588989, 9598949, 9599059, 9599069, 9599969, 9602233, 9605669, 9606599, 9606899, 9608609, 9608969, 9609959, 9611869, 9622259, 9622549, 9625669, 9626299, 9626699, 9642599, 9644009, 9644809, 9644909, 9649909, 9652229, 9654989, 9655559, 9655829, 9656099, 9658049, 9660223, 9660559, 9660659, 9662399, 9664009, 9664909, 9665069, 9665269, 9665609, 9666689, 9668059, 9669929, 9681169, 9688859, 9689249, 9689689, 9694429, 9694589, 9695299, 9695459, 9696059, 9696529, 9696559, 9698069, 9698089, 9699509, 9699959, 9804859, 9808969, 9809659, 9811859, 9818689, 9818881, 9844699, 9846989, 9852529, 9854899, 9854969, 9855229, 9855259, 9855689, 9856409, 9856529, 9858119, 9858809, 9864949, 9865109, 9865589, 9865811, 9865819, 9866669, 9868189, 9869869, 9885611, 9885859, 9885989, 9886559, 9888929, 9894569, 9895229, 9895889, 9896449, 9896489, 9898859, 9900493, 9904333, 9904343, 9904649, 9905459, 9906569, 9909943, 9922559, 9925969, 9926269, 9929999, 9930443, 9932669, 9942659, 9944663, 9949249, 9949663, 9949943, 9949963, 9952469, 9955559, 9956069, 9956459, 9962999, 9964489, 9964859, 9966269, 9966559, 9968809, 9969229, 9969559, 9969629, 9984589, 9986069, 9992699, 9994333, 9994363, 9999299, 9999433, 9999463, 9999943, 10011101, 10045001, 10054001, 10054481, 10111001, 10141111, 10404451, 10444051, 10508051, ...}

This set is currently not known, and might be extremely difficult to found.

Minimal set of emirps in dozenal:

{15, 51, 57, 5B, 75, B5, 107, 117, 11B, 12B, 13B, 16B, 17B, 19B, 1A7, 701, 711, 76B, 7A1, 7BB, B11, B21, B31, B61, B67, B71, B91, BB7, 1011, 1021, 1061, 10B1, 10BB, 1101, 11A1, 1201, 1261, 1297, 1391, 1437, 1467, 14B1, 14BB, 1601, 1621, 1667, 1677, 1681, 1747, 1797, 184B, 1861, 1931, 1937, 1947, 1A11, 1AB1, 1B01, 1B0B, 1B27, 1B41, 1B4B, 1BA1, 5025, 5045, 5095, 5205, 5385, 5405, 5455, 54A5, 5545, 5585, 5835, 5855, 5905, 59A5, 5A45, 5A95, 703B, 7097, 7187, 727B, 72A7, 72B1, 730B, 7341, 7377, 7391, 739B, 7471, 7477, 7491, 749B, 74A7, 7641, 7661, 7687, 7737, 773B, 7747, 7761, 7817, 7867, 789B, 7907, 7921, 794B, 7971, 79AB, 7A27, 7A47, B01B, B037, B0B1, B10B, B18B, B27B, B2AB, B307, B377, B3BB, B481, B497, B4B1, B727, B72B, B7AB, B81B, B937, B947, B987, BA2B, BA7B, BA97, BB01, BB3B, BB41, 1000B, 104A1, 10891, 109A1, 10AAB, 11961, 11981, 12277, 12371, 12627, 12647, 12731, 12787, 12791, 130A1, 13247, 13347, 13371, 13641, 13721, 13A81, 14461, 14471, 14631, 14691, 14781, 14787, 14891, 148A1, 16327, 16441, 166A1, 16837, 16887, 16911, 17267, 17287, 17321, 17331, 17441, 17777, 18397, 18497, 18687, 18741, 18787, 18911, 189A1, 18A31, 18AAB, 18B37, 19641, 19721, 19801, 19841, 19997, 1A031, 1A401, 1A40B, 1A48B, 1A4AB, 1A661, 1A841, 1A901, 1A981, 1B347, 1B8BB, 50885, 52345, 523A5, 52895, 52A85, 53055, 53265, 53565, 53995, 54325, 55035, 55065, 55565, 55595, 55A65, 56055, 56235, 56535, 56555, 56A55, 56AA5, 582A5, 58805, 58A25, 58AA5, 59555, 59825, 59935, 5A285, 5A325, 5AA65, 5AA85, 700AB, 70277, 70437, 70727, 707AB, 7108B, 71647, 7188B, 720B7, 72197, 7222B, 7233B, 72361, 72447, 724B7, 72621, 72677, 72707, 72797, 728B7, 72A4B, 73167, 731AB, 7334B, 73407, 73467, 7347B, 73497, 7378B, 73861, 73A8B, 73B81, 74167, 74231, 7431B, 74331, 74387, 743B1, 74427, 74617, 74621, 74887, 74A4B, 74B97, 76137, 76147, 76271, 76437, 7704B, 77207, 7721B, 77221, 77627, 77697, 77771, 7777B, 77A0B, 77A87, 78271, 78347, 7837B, 783AB, 78681, 78721, 7872B, 78741, 7877B, 78781, 7878B, 78847, 78861, 78A77, 79127, 7923B, 79381, 79437, 79481, 79677, 79727, 7990B, 79991, 79B47, 7A07B, 7A20B, 7A38B, 7A40B, 7A84B, 7A99B, 7AA8B, 7B027, 7B427, 7B827, B0001, B002B, B008B, B024B, B02A7, B039B, B04A1, B04A7, B079B, B098B, B0997, B0A77, B0B6B, B1277, B1347, B1A4B, B200B, B204B, B2227, B228B, B268B, B2787, B283B, B28BB, B294B, B2BBB, B3297, B3327, B346B, B364B, B382B, B387B, B388B, B396B, B398B, B402B, B4077, B420B, B42BB, B4337, B463B, B477B, B48A7, B492B, B4A1B, B4A27, B4A47, B643B, B693B, B6ABB, B6B0B, B70A7, B7387, B7437, B774B, B7777, B7787, B783B, B800B, B8017, B822B, B83A7, B84A1, B862B, B8737, B8787, B8817, B883B, B890B, B893B, B8A37, B8AA7, B8B9B, B930B, B94BB, B970B, B99A7, B9B8B, BA007, BA137, BA387, BA4A1, BA707, BAA01, BAA81, BB24B, BB49B, BB82B, BB8B1, BBA6B, BBB2B, 100031, 103031, 10404B, 1083A1, 111211, 112111, 112411, 112821, 112911, 113811, 113831, 114141, 114181, 114211, 114241, 114911, 118131, 118311, 119211, 119291, 119411, 121381, 121491, 122131, 122181, 122191, 1222A1, 1223A1, 122447, 122921, 122A31, 123281, 123327, 123377, 123381, 123867, 123AA1, 124441, 124991, 124A31, 124AA1, 128211, 128341, 1288A1, 128921, 128981, 128A21, 129221, 129491, 129821, 129991, 12A3A1, 12A821, 12A8A1, 130001, 130301, 130831, 130841, 131181, 131221, 131281, 131811, 132737, 133081, 133A31, 134987, 136287, 136827, 138031, 138311, 138487, 138871, 138881, 139697, 139977, 13A221, 13A331, 13A421, 1400A1, 1400AB, 140A0B, 141161, 141411, 142411, 142A81, 143481, 143821, 143A61, 144421, 14444B, 147277, 147291, 148031, 148777, 14880B, 148A0B, 148B47, 149091, 14A481, 161141, 164827, 164A61, 169691, 16A341, 16A461, 176227, 178831, 178887, 180331, 180A81, 181131, 181221, 181411, 181491, 182131, 182321, 182491, 183121, 183321, 183637, 183877, 184341, 184881, 184A41, 187367, 187627, 188481, 188791, 188831, 188981, 188987, 188A81, 189491, 189821, 189881, 189977, 18A081, 18A241, 18A881, 18A891, 18B997, 190941, 191221, 192741, 192911, 1940A1, 194121, 194181, 194281, 194921, 194981, 194A91, 196961, 197881, 198287, 198767, 198A81, 199421, 199921, 19A491, 1A0041, 1A0491, 1A2221, 1A3221, 1A3801, 1A3A21, 1A8821, 1A88AB, 1A8A21, 1AA321, 1AA421, 1B8887, 1B9777, 1BAABB, 1BABAB, 1BBAAB, 1BBBBB, 500085, 505655, 522365, 524885, 525655, 526595, 526935, 526945, 532395, 5335A5, 536695, 539625, 53A655, 544645, 546365, 546445, 548465, 548495, 548665, 548885, 549295, 549625, 549965, 549985, 552295, 5533A5, 5560A5, 556505, 556525, 556965, 556A35, 559395, 560685, 5608A5, 562995, 563225, 563645, 5636A5, 564845, 5660A5, 566395, 566485, 566845, 566985, 569395, 569655, 569895, 569945, 56A685, 580005, 5806A5, 582485, 584285, 584665, 586065, 586885, 586A65, 588425, 588685, 588845, 589665, 589945, 592255, 592495, 592945, 593235, 593595, 593665, 593955, 593965, 594295, 594845, 595395, 595625, 596635, 598965, 599265, 5A0655, 5A0665, 5A3355, 5A5335, 5A6085, 5A6365, 5A8065, 700067, 70024B, 700387, 700B47, 702267, 70298B, 703047, 703067, 70408B, 7044AB, 70478B, 704A8B, 70700B, 70742B, 70770B, 707AA7, 707B07, 708777, 70A22B, 70A48B, 70A80B, 70A887, 70AB77, 70B387, 70B707, 70B847, 70B887, 71040B, 713637, 714227, 7144B7, 717767, 71B337, 720637, 721247, 7223B7, 722417, 722467, 7224AB, 722637, 722671, 722977, 7229B7, 723137, 7231B7, 723267, 723321, 724337, 726167, 726247, 726781, 727177, 7282AB, 728461, 728631, 72909B, 72910B, 729877, 729887, 72991B, 72993B, 72A0AB, 72A92B, 72B387, 72B977, 731327, 732947, 73321B, 73323B, 73332B, 733427, 73371B, 733A7B, 733B17, 734347, 73482B, 734A3B, 734B37, 736027, 736227, 736317, 736381, 7369A7, 737231, 73882B, 738897, 739AA7, 73A387, 73A42B, 73AA2B, 73AA4B, 73AB37, 73AB87, 73B437, 73B487, 73B997, 73BA37, 740087, 740307, 74042B, 74048B, 74070B, 74078B, 74148B, 74202B, 742127, 742297, 74238B, 742487, 742627, 74323B, 74333B, 743437, 743697, 743B47, 744087, 74408B, 744221, 744497, 74708B, 74842B, 748B07, 749237, 74A02B, 74B007, 74B347, 74B841, 760007, 760307, 760367, 761627, 761697, 761997, 762207, 762327, 763067, 763781, 7639A7, 764227, 766497, 766A77, 7670A7, 767677, 767717, 767891, 768321, 769277, 76A0A7, 770087, 771727, 771777, 771B97, 77220B, 772741, 77282B, 772967, 772997, 773321, 7741AB, 776767, 77700B, 7770A7, 777177, 77718B, 7771AB, 77724B, 77744B, 777807, 777841, 7779B1, 778381, 7784AB, 77882B, 778927, 778A8B, 77900B, 779227, 779797, 77992B, 779931, 779981, 779B27, 77A48B, 77A667, 77A6A7, 77A7A7, 77A9B7, 77B197, 77BA07, 780047, 780077, 78042B, 780447, 78148B, 781AAB, 78220B, 782631, 782891, 7828AB, 783007, 78318B, 783A37, 783B07, 783B27, 78420B, 784247, 78444B, 78474B, 78481B, 784831, 784A2B, 784B37, 784B87, 787A4B, 787A97, 78824B, 78871B, 78874B, 788871, 7888B1, 788927, 788A07, 788AAB, 788B07, 789431, 789881, 78A28B, 78A32B, 78A3B7, 78A83B, 78A8B7, 78B487, 78BA37, 791B77, 79208B, 792247, 7922B7, 793BA7, 794447, 794667, 796167, 796347, 796931, 797977, 797A97, 797AA7, 798837, 79888B, 799167, 799277, 799B37, 799B81, 79A787, 79A797, 79B177, 79BAA7, 7A024B, 7A028B, 7A0767, 7A0777, 7A0A67, 7A422B, 7A423B, 7A6A77, 7A70AB, 7A7A77, 7A7AA7, 7A9367, 7A9637, 7A97B7, 7AA22B, 7AA4AB, 7AA707, 7AA797, 7AA7A7, 7AA937, 7AAB97, 7AB397, 7B1327, 7B2297, 7B3227, 7B3A87, 7B4417, 7B79A7, 7B8A87, 7B9227, 7B9A77, B004BB, B0060B, B00707, B00777, B00977, B00A3B, B01927, B02277, B02287, B02487, B0260B, B0323B, B034AB, B03A3B, B03A6B, B04017, B0403B, B0448B, B044BB, B0466B, B0490B, B049BB, B04B9B, B0600B, B0604B, B0620B, B0642B, B0666B, B0669B, B06A0B, B07047, B0707B, B0738B, B07707, B082BB, B0842B, B0847B, B08841, B0889B, B08A07, B08B4B, B08B8B, B0940B, B09BAB, B0A041, B0A60B, B0A841, B0AB8B, B0ABAB, B0BB9B, B12337, B17337, B17887, B18487, B19927, B1BAAB, B20247, B2066B, B2082B, B2086B, B20A47, B20B2B, B2242B, B224A7, B2263B, B22A07, B22AA7, B23337, B2339B, B2348B, B2362B, B23A87, B24047, B24087, B2422B, B2460B, B24707, B2480B, B24847, B24A37, B24B8B, B2632B, B2634B, B2644B, B2802B, B28277, B28437, B28837, B28877, B29977, B29A27, B2A487, B2AA37, B2B02B, B2B89B, B3038B, B3040B, B308AB, B3230B, B32337, B32347, B324A7, B3269B, B33347, B3444B, B3622B, B3668B, B36AAB, B38A87, B3934B, B3949B, B39927, B3A00B, B3A30B, B3A437, B400BB, B40401, B4049B, B404AB, B4060B, B42007, B420A7, B4236B, B42777, B42887, B4289B, B4347B, B4362B, B4384B, B438AB, B4393B, B4443B, B44441, B44487, B4462B, B44777, B448AB, B4606B, B4628B, B466AB, B46B6B, B47487, B47887, B4834B, B4898B, B48A4B, B4949B, B4A787, B4A84B, B4AA37, B4B80B, B600BB, B6064B, B60A9B, B6246B, B6324B, B6366B, B63AAB, B6426B, B6496B, B6498B, B64A8B, B64AAB, B6602B, B6636B, B6640B, B6660B, B6802B, B690BB, B6946B, B6A30B, B6A39B, B6A68B, B6A6AB, B6B64B, B6B8AB, B6BA8B, B7070B, B7434B, B7480B, B7487B, B7847B, B7A337, B80297, B803AB, B80407, B80447, B80747, B8088B, B80BBB, B81387, B81777, B820A7, B8264B, B82A87, B8303B, B83247, B8370B, B84047, B84147, B84187, B8432B, B8440B, B844BB, B8499B, B849AB, B84A07, B84A77, B8663B, B86A6B, B86BAB, B87047, B87407, B8808B, B88897, B889BB, B89207, B8946B, B8984B, B8A39B, B8A407, B8A46B, B8A877, B8AB6B, B8B42B, B8B80B, B8BA0B, B8BBAB, B90927, B90ABB, B9332B, B933AB, B938AB, B93A6B, B93A8B, B93A9B, B9404B, B940AB, B9469B, B9493B, B9494B, B9623B, B9649B, B9660B, B980BB, B9824B, B9880B, B98A9B, B98B2B, B9948B, B99ABB, B9A06B, B9A0BB, B9A39B, B9A89B, B9AB9B, B9B40B, B9BA9B, B9BB0B, B9BBAB, BA0041, BA049B, BA06BB, BA07A7, BA09BB, BA0A27, BA1477, BA1777, BA2827, BA308B, BA339B, BA34AB, BA404B, BA4227, BA430B, BA43AB, BA4407, BA4877, BA4AA7, BA664B, BA6A6B, BA6AAB, BA803B, BA80BB, BA8287, BA834B, BA839B, BA844B, BA88A1, BA8B6B, BA948B, BAA187, BAA36B, BAA46B, BAA63B, BAA6AB, BAA887, BAAB1B, BAABB1, BAB68B, BAB90B, BABA0B, BABAB1, BABB8B, BABB9B, BABBBB, BB004B, BB006B, BB089B, BB08AB, BB096B, BB0A9B, BB280B, BB400B, BB440B, BB448B, BB60AB, BB8BBB, BB90AB, BB940B, BB988B, BBA09B, BBA99B, BBAAB1, BBB08B, BBB8BB, BBBBAB, BBBBB1, 1000401, 1000A81, 1003041, 1003801, 100888B, 10088A1, 1009401, 1009901, 1030401, 1038441, 1038801, 1038841, 103A441, 1040001, 1040091, 1040301, 1043331, 1048041, 1048331, 1049001, 1083001, 1084831, 1088301, 1088841, 1088AA1, 108A0A1, 108AA81, 1090491, 1094481, 1094881, 1099001, 1099481, 1099881, 10A4441, 10A4491, 10A80A1, 10A8381, 1111191, 1111881, 1113421, 1114191, 1114331, 1114431, 1114481, 1114811, 1114881, 1116141, 1118231, 1118921, 1119241, 1121291, 1122991, 1124341, 1131881, 1132181, 1132331, 1134611, 1142231, 1142921, 1143881, 1144821, 1148321, 1148821, 1161161, 1163161, 1164311, 1166331, 1166631, 1184111, 1184821, 1188331, 1188881, 1194441, 1194941, 1199991, 1213131, 1222281, 1222421, 1222741, 1222867, 1223267, 1223441, 1223637, 1223837, 1224291, 1228281, 1229441, 1229881, 1229AA1, 1232367, 1233131, 1237367, 1238131, 1238411, 1241341, 1242221, 1242281, 1242491, 1243111, 1244427, 1244981, 1248141, 1273227, 1274281, 1282831, 1282847, 1282A81, 1283181, 1283287, 1284411, 1284811, 1284827, 1287227, 1288237, 1288411, 1292411, 1292891, 1292A91, 1294481, 12944A1, 1298111, 1298181, 129A881, 12A8891, 12A9241, 12A9A41, 12AA481, 12AA941, 1313121, 1313321, 1316331, 1318321, 1322281, 1322327, 1322341, 1322411, 1322827, 13228A1, 1323241, 1328111, 1328387, 1328AA1, 132A241, 132AAA1, 1331241, 1331361, 1331441, 1332311, 1333401, 1334111, 1336131, 1336611, 1338341, 1338401, 1338727, 1338811, 133A461, 1341441, 1342AA1, 1343841, 1344111, 1348441, 1362337, 1364497, 1366611, 1378841, 1382781, 1382821, 1383227, 1383AA1, 1384041, 1384381, 13843A1, 1384771, 1384801, 1386927, 138A041, 1397727, 13A2341, 1400841, 1403001, 1403441, 1404491, 140480B, 1404831, 1404A4B, 1408401, 1409081, 1409941, 140A831, 1416111, 1418421, 1421331, 14224A1, 1423231, 14244A1, 1424981, 1429111, 1429A21, 142A231, 142A2A1, 1431421, 1432231, 14323A1, 1432A31, 1432AA1, 1433161, 1434211, 1438331, 1438441, 1440991, 1441331, 1441431, 1442381, 1442A41, 1443041, 1443221, 14434A1, 1444911, 1444941, 1444A01, 1448301, 1448341, 1448431, 1449221, 1449727, 144A0A1, 144A301, 1461661, 1466161, 146AAA1, 1472221, 1472271, 1480041, 1480881, 14808AB, 1483431, 1484081, 1484381, 1484477, 1487271, 1488227, 1488301, 1488427, 1488477, 1488731, 1488801, 1491881, 14922A1, 1494441, 1494911, 1499041, 1499A41, 149AA21, 14A2441, 14A9941, 14A9A21, 1611611, 1613341, 1613611, 1616641, 1623337, 1631331, 1641991, 1649961, 164A331, 1661641, 1692287, 1699461, 169A9A1, 1722291, 1722741, 1724381, 1727281, 1727841, 1733327, 1733827, 1774831, 1782227, 1783237, 18000A1, 1800A8B, 1804841, 1809041, 180A80B, 1812311, 1812891, 1813821, 1818921, 1822221, 1822231, 1822367, 1822421, 1823387, 1823837, 1824481, 1824721, 1827271, 1827327, 1828191, 1828221, 1828387, 1828637, 1828767, 1830481, 18323A1, 1832441, 1832881, 1834271, 1834831, 1834841, 1837767, 1837781, 1838447, 1838A01, 1838AA1, 1840381, 18423A1, 1842777, 1844111, 1844281, 1844901, 1844921, 1847791, 1848277, 1849901, 184AA21, 1872337, 1872831, 1877381, 1880841, 1881111, 1881311, 1881941, 1882191, 1882381, 18824A1, 1883411, 1884111, 1884901, 188800B, 1888277, 1888287, 1888811, 1888977, 1888B87, 1889221, 1889697, 1889901, 188A921, 188AA91, 1890991, 1894241, 1894421, 1898697, 18A0001, 18A2821, 18AA2A1, 18AA801, 18AA8A1, 18AB88B, 18B88AB, 1900401, 1911111, 1911991, 1912881, 1914111, 1918281, 1921211, 1922227, 1922271, 1924221, 1928867, 192AA91, 1940901, 1942421, 1944041, 1944A01, 19496A1, 1968227, 1969A91, 1977481, 1982181, 1982921, 1988A21, 198A2A1, 1990441, 1990981, 1991191, 1991461, 1992211, 1992867, 1998727, 1999911, 199AAA1, 19A2921, 19A2AA1, 19A9691, 19AA291, 19AA881, 19AAAA1, 1A00081, 1A08A01, 1A0A441, 1A0A801, 1A22941, 1A28AA1, 1A2A241, 1A2A891, 1A2AA81, 1A32341, 1A32381, 1A32481, 1A34831, 1A42241, 1A42881, 1A43441, 1A44241, 1A44921, 1A69491, 1A82231, 1A88001, 1A8808B, 1A8AA81, 1A9A961, 1AA2341, 1AA2431, 1AA2A91, 1AA3831, 1AA80AB, 1AA8231, 1AA82A1, 1AA8381, 1AA8801, 1AA880B, 1AA9221, 1AAA231, 1AAA641, 1AAA991, 1AAAA91, 1AAB88B, 1AABBAB, 1AB8A8B, 1ABAB8B, 1B48487, 1B83337, ...}

This set is currently not known, and might be extremely difficult to found.

Minimal set of totients in decimal:

{1, 2, 4, 6, 8, 30, 70, 500, 900, 990, 5590, 9550, 555555555550}

This set is complete, reference: http://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf

Minimal set of totients in dozenal:

{1, 2, 4, 6, 8, A, 30, 50, 70, 90, B0}

This set is complete, since any remain number must end with 0 and only 3, 5, 7, 9, B can be in front of it, but all of 30, 50, 70, 90, B0 are already in this set.

Minimal set of "totients plus 1" in decimal:

{2, 3, 5, 7, 9, 11, 41, 61, 81}

This set is complete, reference: http://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf

Minimal set of "totients plus 1" in dozenal:

{2, 3, 5, 7, 9, B, 11, 41, 61, 81, A1}

This set is complete, since any remain number must end with 1 and only 1, 4, 6, 8, A can be in front of it, but all of 11, 41, 61, 81, A1 are already in this set.

Minimal set of "totients plus 2" in decimal:

{3, 4, 6, 8, 10, 12, 20, 22, 50, 72, 90, 770, 992, 5592, 9552, 555555555552}

This set is complete if and only if there are no totients of the form 6{9}8, and such totients are conjectured not to exist, reference: http://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf

Minimal set of "totients plus 2" in dozenal:

{3, 4, 6, 8, A, 10, 12, 20, 22, 50, 52, 70, 72, 90, 92, B0, B2}

This set is complete.

Minimal set of "totients plus 3" in decimal:

{4, 5, 7, 9, 11, 13, 21, 23, 31, 33, 61, 63, 81, 83}

This set is complete, reference: http://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf

Minimal set of "totients plus 3" in dozenal:

{4, 5, 7, 9, B, 11, 13, 21, 23, 33, 61, 63, 83, A3, 301, 801, A01, AA1, 3A81, 8A81, 38881, 88881}

This set is complete.

Minimal set of "totients plus 4" in decimal:

{5, 6, 8, 10, 12, 14, 20, 22, 24, 32, 34, 40, 44, 70, 74, 92, 300, 472, 772, 900, 904, 994}

This set is complete if and only if there are no totients of the form {3,9}26 or {3,9}86, and such totients are conjectured not to exist, reference: http://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf

Minimal set of "totients plus 4" in dozenal:

{5, 6, 8, A, 10, 12, 14, 20, 22, 24, 30, 34, 40, 42, 44, 70, 72, 74, 90, 92, 94, B0, B2, B4, 332}

This set is complete.

Minimal set of "totients minus 1" in decimal (0 is not counted):

{1, 3, 5, 7, 9}

This set is very easy to proven to be complete, since any such number ends with 1, 3, 5, 7, or 9.

Minimal set of "totients minus 1" in dozenal (0 is not counted):

{1, 3, 5, 7, 9, B}

This set is very easy to proven to be complete, since any such number ends with 1, 3, 5, 7, 9, or B.

Minimal set of "totients minus 2" in decimal (0 is not counted):

{2, 4, 6, 8, 10, 30, 50, 70, 90}

This set is complete.

Minimal set of "totients minus 2" in dozenal (0 is not counted):

{2, 4, 6, 8, A, 90, 1300, 3B30, 133130}

This set is complete if and only if there are no totients of the forms 1{0}2, 3{0}2, 5{0}2, 7{0}2, B{0}2, {1}2, {3}2, {5}2, {7}2, {B}2, and such totients are conjectured not to exist, as there are only few totients end with 2, for such numbers, see https://oeis.org/A063668

Minimal set of "totients minus 3" in decimal:

{1, 3, 5, 7, 9}

This set is very easy to proven to be complete, since any such number ends with 1, 3, 5, 7, or 9.

Minimal set of "totients minus 3" in dozenal:

{1, 3, 5, 7, 9, 8B}

This set is complete if and only if there are no totients of the form 1{0}2, {0,2,4,6,8,A}12, {0,2,4,6,8,A}32, {0,2,4,6,8,A}52, {0,2,4,6,8,A}72, {0,2,4,6,8,A}B2, and such totients are conjectured not to exist, as there are only few totients end with 2, for such numbers, see https://oeis.org/A063668

Minimal set of "totients minus 4" in decimal (0 is not counted):

{2, 4, 6, 8, 50, 100, 700, 900, 3300, 300000}

This set is complete if and only if there are no totients of the form {1}4, {3}4, {7}4, {9}4, and such totients are conjectured not to exist.

Minimal set of "totients minus 4" in dozenal (0 is not counted):

{2, 4, 6, 8, 10, 30, 50, 70, 90, B0, 35A}

This set is complete if and only if there are no totients of the form {1,3,5,7,9,B}02, {1,3,5,7,9,B}22, {1,3,5,7,9,B}42, {1,5,7,9,B}62, {1,3,5,7,9,B}82, {1,3,5,7,9,B}A2, and such totients are conjectured not to exist, as there are only few totients end with 2, for such numbers, see https://oeis.org/A063668

Minimal set of non-single-digit totients in decimal:

{10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 276, 500, 776, 876, 900, 904, 990, 5590, 9550, 555555555550}

I think that this set is complete.

Minimal set of non-single-digit totients in dozenal:

{10, 14, 16, 18, 1A, 20, 24, 26, 28, 30, 34, 36, 38, 3A, 40, 44, 46, 48, 4A, 50, 54, 56, 5A, 60, 66, 68, 6A, 70, 74, 78, 80, 84, 86, 88, 8A, 90, 92, 94, 98, A0, A6, A8, AA, B0, B4, B6, B8, 27A, 2BA, 77A, 796, 7BA, 9BA, B2A, 222A, 229A, 299A, 3B32, 729A, 9776, 997A, 9996, B97A, 7999A, 9999A, A5B32, BB79A, BBB7A, 133132, 372532, 821732, B1B232}

I think that this set is complete.

Minimal set of range of Dedekind psi function in decimal:

{1, 3, 4, 6, 8, 20, 72, 90, 222, 252, 500, 522, 552, 570, 592, 750, 770, 992, 7000, 5555555555555555555555555555555555555555555555555555555555555555555550}

This set is complete, reference: https://arxiv.org/pdf/1607.01548.pdf

Minimal set of range of Dedekind psi function in dozenal:

{1, 3, 4, 6, 8, 20, 50, 52, 70, 90, 92, A0, B0, 222, 272, 2A2, 2B2, 722, 772, 7A2, B22, B72, 7BBB2, A07B2, A7BB2, AA7B2, AAAA2, AABB2, BAAA2, BBBB2, ABAAB2, ABBAB2, ABBBA2, BABAB2, BBAAB2, BBBAA2}

I think that this set is complete.

Minimal set of range of sigma function in decimal:

{1, 3, 4, 6, 7, 8, 20, 90, 222, 252, 255, 500, 522, 552, 592, 952, 992, 5555555555555555555555555555555555555555555555555555555555555555555550}

I think that this set is complete.

Minimal set of range of sigma function in dozenal:

{1, 3, 4, 6, 7, 8, 20, 50, 52, 90, 92, A0, B0, 222, 2A2, 2B2, B22, 2BA9}

I think that this set is complete.

Minimal set of Fibonacci numbers in decimal (0 is not counted as Fibonacci number):

{1, 2, 3, 5, 8}

This set is conjectured to be complete, but not proven.

Minimal set of Fibonacci numbers in dozenal (0 is not counted as Fibonacci number):

{1, 2, 3, 5, 8, 47}

This set is conjectured to be complete, but not proven, any remain numbers must end with 00, B9, or BB, since Fibonacci numbers end with 00, 01, 02, 03, 05, 08, 11, 19, 2A, 47, 75, A3, B4, B9, BB.

Minimal set of Lucas numbers in decimal (2 is not counted as Lucas number):

{1, 3, 4, 7, 29, 521}

This set is conjectured to be complete, but not proven.

Minimal set of Lucas numbers in dozenal (2 is not counted as Lucas number):

{1, 3, 4, 7, B, 25, 22A}

This set is conjectured to be complete, but not proven.

Last fiddled with by sweety439 on 2021-02-23 at 12:50
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Old 2021-02-22, 13:49   #123
sweety439
 
Nov 2016

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Quote:
Originally Posted by sweety439 View Post
Reserve this family to n=100K

Update sieve files.
At n=92489, no (probable) prime found.
Attached Files
File Type: txt base 40 SQd family status.txt (14.3 KB, 11 views)
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Old 2021-02-22, 16:42   #124
sweety439
 
Nov 2016

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Quote:
Originally Posted by sweety439 View Post
Although some such minimal sets are not known, and might be extremely difficult to found (these sets may be more difficult to found in larger bases, such as bases 17 through 36), the basic theorem of minimal sets says that the minimal set is always finite. However, finding the number of elements in these minimal sets is very difficult.
For the number of elements in the minimal set of some sets, such as squares (1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276 ...) and powers of 2 ({1, 2, 4, 8, 65536, ...), even no upper bounds are known (only lower bounds are known, the lower bounds for these two sets are 55 and 5, respectively), but for the number of elements in the minimal set of primes in base b, there is a known way to find the upper bound, even if b is large.

This is the data for the lower bound and the upper bound for the number of elements in the minimal set of primes for bases 2<=b<=50: (if one allows probable primes in place of proven primes) (only cases for bases 2<=b<=16 and b = 18, 20, 22, 23, 24, 30, 42 are proven, and bases 13, 23 need primality proving of probable primes)

Code:
n,lower bound,upper bound
2,2,2
3,3,3
4,3,3
5,8,8
6,7,7
7,9,9
8,15,15
9,12,12
10,26,26
11,152,152
12,17,17
13,228,228
14,240,240
15,100,100
16,483,483
17,1279,1280
18,50,50
19,3462,3463
20,651,651
21,2600,2601
22,1242,1242
23,6021,6021
24,306,306
25,17597,17609
26,5662,5664
27,17210,17215
28,5783,5784
29,57283,57297
30,220,220
31,79182,79206
32,45205,45283
33,57676,57709
34,56457,56490
35,182378,182393
36,6296,6297
37,314988,315263
38,106838,106915
39,230317,230360
40,37773,37774
41,689061,689396
42,4551,4551
43,900795,901331
44,255911,256014
45,323437,323484
46,399012,399125
47,1436289,1437283
48,29103,29109
49,4365269,4366452
50,189914,189976
(there are some known (probable) primes not in data in https://github.com/curtisbright/mepn...ee/master/data and https://github.com/RaymondDevillers/primes: P81993SZ in base 36 (found by me), FYa22021 in base 37 (found by CRUS Riesel problem base 37), R8a20895 in base 37 (found by CRUS Riesel problem base 37), QaU12380X in base 40 (found by me), O0185211 in base 45 (found by CRUS Sierpinski problem base 45), 11c0297361 in base 49 (found by CRUS Sierpinski problem base 49), Fd0183401 in base 49 (found by CRUS Sierpinski problem base 49), SLm52698 in base 49 (found by CRUS Riesel problem base 49), Ydm16337 in base 49 (found by CRUS Riesel problem base 49), thus these primes can be added to the lower bounds of corresponding bases, and the corresponding families can be removed from the unsolved families, however, the families O{0}1F1 in base 45, O{0}ZZ1 in base 45, S6L{m} in base 49, YUUd{m} in base 49, YUd{m} in base 49, are not tested for all numbers which are not contain the known minimal primes O0185211 in base 45, SLm52698 in base 49, Ydm16337 in base 49, as subsequence, since they are only tested to length 10000, thus these families might have minimal primes, and we cannot definitely say that these five families can be removed, unless we tested: O{0}1F1 in base 45, O{0}ZZ1 in base 45, to 18520 0's with no (probable) prime found; S6L{m} in base 49, to 52697 m's with no prime found; YUUd{m} in base 49, YUd{m} in base 49, to 16336 m's with no prime found. (however, the family AO{0}1 in base 45 is known to be prime at minimal AO0447901 (found by CRUS Sierpinski problem base 45), thus this family contains no minimal prime, and we can definitely say that this family can be removed)

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Old 2021-02-23, 04:28   #125
LaurV
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Random question:

Quote:
Originally Posted by sweety439 View Post
Minimal set of emirps in decimal:
{13, 17, 31, 37, 71, 73, 79, 97, 149, <blah blah>...
Why is 11 not in the set? It would have changed a lot of things after...
edit: same for dozenal... 1112=1310, a prime... in fact, it seems there are other mistakes in those sets too..
edit 2: man, use those freaking code tags, please! (ideally it would be if you stop posting those long strings of meaningless numbers at all, but doh, I can't force you to, as long as you stay inside of your blog, but please have mercy of our screen space, otherwise I promise you I would lobby for a longer ban!).

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Old 2021-02-23, 12:42   #126
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Quote:
Originally Posted by LaurV View Post
Random question:


Why is 11 not in the set? It would have changed a lot of things after...
edit: same for dozenal... 1112=1310, a prime... in fact, it seems there are other mistakes in those sets too..
edit 2: man, use those freaking code tags, please! (ideally it would be if you stop posting those long strings of meaningless numbers at all, but doh, I can't force you to, as long as you stay inside of your blog, but please have mercy of our screen space, otherwise I promise you I would lobby for a longer ban!).
See https://primes.utm.edu/glossary/page.php?sort=emirp and https://oeis.org/A006567, palindromic primes are not emirps, emirps are primes whose reversal is a DIFFERENT prime.

Also, both posts #119 and #122 changed to code things.

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Old 2021-02-23, 14:34   #127
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Ok. It sounds right. Sorry, I had it in my mind that the mirrored number has to be prime, but I didn't know that it has to be a different prime. Still don't know why, and I assume the definition is arbitrary, anyhow. It doesn't seem to have any mathematical importance. But if they are so defined, then they are so defined.

Thanks for adding code tags to those posts.
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Old 2021-02-24, 02:35   #128
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Quote:
Originally Posted by sweety439 View Post
At n=92489, no (probable) prime found.
Base 40 S{Q}d family (86*40^n+37)/3 tested to n=100000, no (probable) prime found, base released.
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File Type: txt base 40 SQd family status.txt (24.0 KB, 6 views)
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