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balanced prime  and  pernicious n.

First numbers that belong to both sets:

5, 157, 173, 211, 257, 563, 607, 653, 733, 947, 1123, 1187, 1367, 1747, 1753, 2677, 2963, 3313, 3637, 3733, 4657, 4993, 5393, 6317, 6323, 6373, 7841, 9397, 10253, 10853, 11411, 11903, 12547, 12583, 14753, 15803, 16427, 16567, 16619, 16937, 16987, 17047, 17477, 18217, 18433, 18731, 19483, 19603, 20161, 20347, 20521, 22051, 22447, 24097, 24413, 26723, 28597, 28927, 29599, 30637, 30643, 31327, 32479, 32497, 32993, 34877, 35129, 35911, 36229, 39113, 39863, 40063, 40933, 40961, 42209, 42533, 44519, 44543, 46147, 48023, 49633, 52957, 54623, 54773, 55207, 57853, 58129, 58237, 59393, 62207, 63493, 63703, 64627, 65413, 67619, 67783, 68099, 68483, 68743, 71843, 72481, 74317, 76561, 76673, 77023, 77557, 79867, 80251, 81439, 81749, 84053, 84137, 87517, 89003, 91297, 91387, 92227, 93083, 95917, 95923, 96431, 96763, 97967, 99823, 100937, 103177, 104953, 105367, 105373, 106297, 108439, 108923, 110459, 110603, 111611, 113963, 114083, 114473, 115337, 116719, 117437, 117757, 121007, 121327, 121447, 123887, 125441, 127643, 127843, 129209, 130483, 130693, 131041, 133103, 133571, 133717, 134639, 135301, 137321, 137393, 138107, 141073, 141263, 143159, 144247, 144253, 145213, 145661, 145879, 146843, 147347, 148633, 148867, 149027, 149497, 150083, 151163, 151477, 151637, 151687, 156241, 159623, 163211, 163847, 163853, 165443, 166949, 167759, 168737, 168893, 169199, 169783, 169837, 170081, 171923, 172433, 172987, 173183, 173561, 176401, 176983, 177217, 177487, 179807, 179813, 181439, 182279, 182593, 183361, 184321, 184837, 185063, 185267, 186013, 186253, 186271, 186877, 187367, 187373, 187699, 188317, 188437, 188933, 189473, 189983, 191467, 192743, 192991, 194653, 195241, 195407, 195913, 195997, 196187, 196739, 197683, 197773, 201151, 201809, 202999, 203017, 203227, 204013, 205957, 206203, 206917, 209743, 210631, 211193, 212677, 212903, 214657, 216731, 217069, 221233, 222947, 223481, 224221, 224467, 225983, 226637, 226643, 227153, 227419, 227581, 228587, 228677, 234287, 234727, 237277, 237331, 237821, 238451, 239423, 240733, 241327, 241951, 244021, 245527, 245771.

Some larger examples :