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good prime  and  happy n.

First numbers that belong to both sets:

97, 331, 563, 739, 937, 1009, 1151, 1277, 1427, 1847, 2063, 2333, 2339, 3907, 4217, 4441, 4447, 4481, 4871, 6037, 6197, 6637, 7451, 7457, 8081, 8521, 8597, 9103, 9133, 10067, 10847, 12473, 13679, 14321, 14387, 14407, 16417, 16811, 17317, 17827, 18899, 19421, 19889, 20717, 21481, 21491, 23741, 25111, 25561, 26669, 27397, 27611, 28387, 29833, 29983, 30637, 30803, 36433, 38593, 38669, 40471, 40751, 41011, 41113, 41131, 41507, 44491, 46141, 47041, 47051, 47111, 48311, 48397, 48479, 48731, 48757, 48973, 50741, 50821, 51407, 52529, 56359, 61441, 64997, 66337, 67033, 67121, 69761, 69809, 69991, 70117, 70841, 73517, 73999, 74047, 74143, 74699, 77261, 79273, 79601, 80141, 80147, 80471, 80909, 84121, 84947, 86111, 88651, 92297, 94397, 100333, 101267, 103903, 103967, 104459, 106181, 107693, 107827, 110419, 110431, 111409, 112757, 113011, 114593, 115727, 115769, 119657, 120823, 131671, 132229, 134033, 135173, 135347, 136189, 143461, 145391, 148331, 149011, 157037, 159617, 159671, 160541, 160579, 162523, 166147, 168977, 176201, 177883, 181693, 184829, 188677, 191413, 200771, 201473, 201491, 201767, 206177, 206779, 207187, 216641, 216731, 217157, 217163, 218077, 219277, 219353, 224863, 227977, 228197, 229123, 237971, 238093, 240701, 243461, 246119, 250681, 254623, 254869, 263009, 263063, 266599, 266633, 266641, 267131, 267187, 278459, 278479, 278543, 279541, 288179, 288907, 295693, 298559, 298993, 300109, 300137, 311537, 315407, 316189, 316633, 318173, 323333, 324011, 327401, 336029, 339991, 344161, 345461, 361217, 362851, 369959, 371057, 371069, 371131, 371831, 371927, 372611, 373963, 374677, 388109, 388879, 404941, 404951, 404969, 414049, 414677, 429101, 429181, 430499, 431801, 439409, 448519, 453553, 467417, 467471, 467587, 467611, 486023, 486041, 487381, 487649, 487681, 492377, 492587, 493109, 494341, 503227, 503771, 504787, 505027, 512497, 516161, 516947, 524789, 535487, 540149, 541193, 545429, 547087, 556021, 556159, 568987, 569417, 569599, 569711, 579497, 579517, 584279, 584357, 592019, 600203, 601759, 603719, 607931, 610541, 611801, 614477, 615253, 615337, 629683.

Some larger examples :