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good prime  and  self n.

First numbers that belong to both sets:

5, 53, 97, 569, 727, 929, 1087, 2683, 3449, 6037, 6823, 8039, 8219, 8623, 8689, 9319, 10133, 14293, 14519, 15217, 16811, 16879, 19421, 19961, 21313, 22619, 25117, 27611, 30089, 31147, 33329, 35801, 36433, 36523, 40483, 40819, 41113, 46399, 47119, 48311, 50833, 51193, 51329, 51419, 54319, 55579, 64601, 64849, 64871, 66491, 66851, 67033, 67121, 70117, 73939, 74143, 75133, 77471, 80599, 86111, 91939, 92143, 93851, 98867, 100267, 100357, 106273, 107599, 108791, 108881, 110527, 111427, 112909, 120539, 127579, 128321, 128389, 132619, 136333, 139301, 143443, 148123, 156671, 156941, 157007, 159617, 162523, 165311, 166841, 168887, 168977, 175211, 175391, 176201, 177887, 181871, 189139, 195731, 197059, 201449, 206779, 207433, 229123, 236681, 237851, 241229, 243073, 244243, 246469, 253243, 253423, 253601, 254927, 261971, 264739, 274661, 275897, 279407, 288049, 295831, 299881, 302563, 303283, 313543, 317321, 317591, 327401, 333253, 336491, 340519, 345379, 350711, 357551, 360769, 362927, 362951, 369947, 370399, 371233, 371927, 374987, 382549, 387677, 388891, 407119, 417491, 422689, 429101, 431801, 449149, 462401, 462491, 467081, 467417, 486023, 492617, 512419, 535099, 535481, 540119, 547361, 555931, 556021, 556159, 567631, 567857, 568891, 578297, 580549, 598613, 600217, 600283, 600823, 606829, 607931, 608269, 611801, 629687, 633739, 644089, 652189, 663529, 678533, 683687, 686723, 691631, 701219, 703849, 707627, 719149, 723043, 744371, 745517, 745697, 747811, 751123, 757151, 757241, 759287, 763339, 775937, 776117, 776813, 776837, 786691, 786803, 787973, 790271, 792023, 793487, 803389, 819229, 819251, 819407, 819431, 822221, 837923, 846341, 846721, 860291, 875297, 877771, 877837, 885713, 904661, 906259, 916831, 931727, 939737, 949951, 956903, 957937, 960931, 960977, 961427, 980417, 985277, 985433, 996841, 996953, 997019, 1000829, 1005217, 1005911, 1055771, 1055839, 1062343, 1074223, 1092827, 1099487, 1102147, 1128223, 1146671, 1159787, 1159967, 1160129, 1173917, 1174211, 1175561, 1177427, 1179251, 1234511, 1238381, 1243309, 1243939, 1245379, 1253951, 1262411, 1279013, 1281649, 1284169, 1285021, 1294967, 1298611, 1308491, 1309549, 1309661, 1316209, 1316479, 1316591.

Some larger examples :