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good prime  and  twin primes

First numbers that belong to both sets:

5, 11, 17, 29, 41, 59, 71, 101, 149, 179, 191, 227, 269, 311, 347, 419, 431, 569, 599, 641, 809, 821, 1031, 1151, 1277, 1427, 1871, 1997, 2237, 2267, 2339, 2657, 2687, 2999, 3167, 3251, 3299, 3461, 3527, 3821, 4001, 4217, 4229, 4481, 4637, 4931, 4967, 5639, 5651, 6197, 6551, 6659, 6689, 6761, 6779, 6947, 7457, 8087, 8219, 8597, 8999, 10007, 10037, 10067, 10091, 11057, 11699, 11777, 12239, 12377, 13679, 14321, 14387, 15269, 16829, 16979, 17291, 17837, 17909, 18911, 19139, 19379, 19421, 19889, 19961, 20717, 20897, 20981, 21377, 21491, 21557, 22541, 22619, 22637, 22697, 22961, 23537, 23741, 24917, 25301, 25577, 26681, 27407, 27689, 27737, 28277, 28571, 30011, 30089, 30491, 31121, 32027, 32057, 32297, 32969, 33287, 33329, 35729, 35801, 36467, 36527, 38651, 38669, 40427, 40697, 41141, 41177, 41849, 43577, 43889, 43961, 46049, 46091, 46439, 48311, 48407, 48479, 48731, 48989, 51131, 51197, 51341, 51419, 52541, 53591, 54401, 55619, 55661, 61331, 61469, 63311, 64577, 64877, 65027, 65099, 65519, 65537, 66359, 66851, 69737, 69761, 69827, 70001, 70841, 73361, 74159, 75167, 77237, 77261, 77417, 77477, 79811, 80147, 80447, 80471, 80669, 80909, 84179, 84389, 85427, 86111, 88589, 88799, 89519, 91079, 91127, 92177, 92219, 92567, 92639, 93809, 93887, 94307, 94397, 97367, 98297, 98711, 98867, 101111, 103289, 103391, 103967, 103979, 104471, 105227, 106187, 107837, 108707, 108791, 108881, 110477, 112757, 114599, 114641, 115769, 116789, 119549, 119657, 127031, 127217, 127247, 128111, 128201, 128339, 134837, 135347, 135389, 139301, 144161, 148061, 148151, 148859, 149057, 150767, 150881, 156941, 157049, 159671, 160619, 162389, 165311, 166601, 166841, 168449, 168599, 169217, 175391, 175631, 175937, 176327, 177677, 177887, 178247, 178439, 181199, 181667, 181871, 181889, 184487, 184607, 184829, 185531, 189149, 189347, 190577, 191447, 194681, 195047, 195731, 196499, 196991, 200867, 201401, 201449, 201491, 201767, 204299, 205031, 205397, 205949.

Some larger examples :