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balanced prime  and  Chen prime

First numbers that belong to both sets:

5, 53, 157, 211, 257, 263, 563, 653, 947, 977, 1187, 1367, 1511, 1907, 2417, 2963, 3307, 3637, 4409, 4657, 5303, 5387, 5807, 6317, 6863, 6977, 8117, 8747, 10607, 11299, 11411, 11497, 11731, 11903, 12497, 12653, 13043, 13457, 13463, 14543, 15161, 15467, 15767, 15907, 16097, 16427, 16487, 16619, 16787, 16937, 17047, 18217, 19577, 19739, 19949, 20107, 20201, 20521, 20731, 21911, 21997, 22051, 22259, 23327, 23801, 25391, 26041, 26177, 26183, 26393, 26987, 28597, 28927, 29333, 30059, 30449, 30977, 32993, 34267, 34631, 34877, 35753, 35911, 36677, 36821, 37957, 38867, 39113, 42683, 44519, 44741, 45497, 46147, 46889, 47303, 48259, 50417, 50441, 51461, 51551, 52529, 52727, 54443, 54851, 55793, 56087, 57119, 58067, 60337, 60509, 60617, 61637, 62207, 62597, 62633, 63559, 65957, 66431, 66809, 67169, 67511, 67619, 68483, 71011, 71843, 74471, 74489, 74567, 75217, 75527, 75533, 75641, 76913, 77081, 77951, 78797, 78803, 80329, 81077, 81689, 81749, 82463, 82787, 84053, 84137, 84437, 84443, 85049, 85121, 86981, 87517, 89387, 92003, 93083, 94427, 95267, 95917, 96377, 96431, 97967, 98597, 100937, 101483, 104161, 104207, 105977, 108217, 110681, 111509, 111521, 111611, 112181, 113921, 114083, 115337, 116797, 119267, 120557, 120563, 121013, 121447, 122099, 122579, 123887, 126211, 126317, 129209, 130829, 131933, 132511, 132929, 133571, 134639, 135089, 135893, 137321, 137393, 138107, 139079, 139703, 142019, 142427, 143159, 143483, 145037, 145661, 146197, 147347, 149027, 149867, 150407, 153443, 154067, 154291, 155821, 156241, 160829, 161977, 162257, 163211, 163853, 164279, 166949, 167729, 169283, 169313, 170081, 170767, 171329, 171571, 172433, 172877, 173839, 174749, 179167, 179369, 179813, 179969, 182261, 182549, 183361, 183383, 183587, 184321, 184517, 184837, 185057, 185063, 186113, 187877, 188159, 188261, 188317, 188687, 188779, 189983, 191467, 192097, 192161, 193757, 195407, 196187, 197927, 198571, 200177, 200357, 201611, 201809, 202231, 202841, 202859, 206027, 211193, 212099, 214457, 215261, 216113, 218453, 218611, 219071.

Some larger examples :