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balanced prime  and  self n.

First numbers that belong to both sets:

5, 53, 211, 1223, 3313, 5113, 5563, 6373, 12583, 15193, 18433, 20123, 20347, 22259, 30313, 30449, 31327, 34543, 35129, 40933, 45433, 53623, 61723, 63073, 71353, 78259, 78797, 85121, 91243, 97967, 98597, 104113, 105373, 110459, 113921, 121327, 122833, 130483, 132911, 134263, 137321, 144253, 144611, 152623, 160117, 173839, 184727, 187877, 188261, 188687, 188933, 202999, 203227, 218143, 218749, 230227, 240733, 245771, 251353, 251533, 252163, 258551, 258977, 265123, 267797, 284539, 287327, 298847, 303889, 306253, 307669, 310237, 315949, 320113, 321193, 324523, 327017, 329717, 332779, 333433, 341623, 346751, 358607, 358697, 368171, 370399, 370511, 374741, 381401, 385967, 392629, 397217, 408223, 414521, 415153, 418459, 418751, 432343, 438551, 452233, 456439, 470501, 478937, 480761, 482627, 485417, 488003, 488633, 488723, 494933, 496487, 497297, 499027, 501037, 508931, 511001, 537169, 550691, 551569, 555349, 559577, 565361, 567811, 572449, 572807, 576677, 584081, 585587, 591887, 593507, 596611, 604063, 604243, 608471, 608831, 634211, 647327, 647441, 658913, 669937, 670279, 673787, 676129, 676937, 678871, 683597, 683687, 695567, 696623, 697973, 700319, 700633, 702523, 703849, 707717, 712939, 713233, 715817, 726527, 729991, 732311, 747991, 748039, 759131, 772697, 774161, 777737, 789121, 790831, 795533, 797131, 797851, 798257, 800083, 802511, 813401, 813559, 819811, 827821, 830861, 842987, 843079, 848297, 853091, 854957, 855857, 856847, 864341, 865307, 868793, 878783, 883061, 890821, 896537, 904727, 906707, 908597, 909767, 912941, 914629, 919129, 926239, 927677, 929141, 931639, 931907, 936407, 937801, 940727, 946801, 948287, 949973, 954923, 955217, 956993, 964577, 969503, 973801, 976501, 978773, 981049, 986581, 987593, 992141, 999541, 1001279, 1002517, 1013833, 1016959, 1021753, 1037929, 1049201, 1055839, 1061129, 1061623, 1077191, 1091053, 1106509, 1109749, 1111169, 1112159, 1127209, 1129459, 1129729, 1130293, 1132633, 1135333, 1140709, 1143679, 1145411, 1145479, 1146043, 1154233, 1154771, 1157381, 1159271, 1164829, 1175717, 1192817, 1197947, 1200943, 1213027, 1217927, 1221463, 1222723, 1222789, 1231721, 1239551, 1243691, 1250443, 1255123, 1256897, 1265987, 1266731, 1267429, 1279667, 1281739, 1282751, 1288823, 1289003.

Some larger examples :