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balanced prime  and  evil n.

First numbers that belong to both sets:

5, 53, 257, 263, 373, 593, 977, 1103, 1223, 1511, 1907, 2287, 2417, 2903, 3307, 4409, 4457, 4597, 4691, 5303, 5387, 6263, 6977, 7523, 7583, 8117, 8713, 8747, 9473, 9871, 10657, 11299, 11497, 12497, 12647, 12653, 12841, 12973, 13457, 13463, 15767, 15797, 15907, 15913, 16223, 16487, 16787, 17431, 17483, 18341, 19463, 19477, 19577, 19739, 19949, 20107, 21163, 21661, 21997, 22073, 22259, 23327, 23333, 23767, 23801, 24077, 24677, 25463, 25621, 25673, 26177, 26183, 26393, 26693, 27067, 27773, 29173, 29269, 29333, 29873, 30059, 30097, 30103, 30449, 30977, 32083, 33113, 33857, 34543, 34631, 34673, 35753, 35851, 36067, 36677, 36821, 37957, 38183, 38867, 40111, 40283, 41017, 41263, 41479, 41513, 41603, 41947, 42187, 42683, 43189, 44041, 45757, 47527, 47869, 48259, 48533, 49069, 50417, 50441, 51413, 51461, 51551, 52529, 52727, 53617, 53623, 54673, 54851, 55793, 55843, 56087, 56093, 56149, 56437, 56773, 56957, 57119, 58543, 58693, 59113, 60083, 60133, 60337, 60509, 61553, 61637, 62213, 62597, 63907, 65557, 65707, 65957, 66431, 66809, 66883, 67169, 67511, 68699, 69623, 71011, 71993, 72031, 72733, 73127, 73477, 73523, 74567, 74897, 75217, 75527, 75533, 75641, 75787, 76579, 76913, 77081, 77563, 77951, 78797, 78803, 79823, 79973, 80917, 81077, 81689, 81937, 82463, 82493, 82787, 82793, 83003, 83437, 83443, 84443, 85049, 85121, 85837, 86137, 86263, 87427, 87547, 89107, 89113, 89387, 89527, 89983, 92003, 93053, 93377, 94427, 94433, 95267, 95273, 95707, 96001, 96377, 98473, 98597, 101287, 103669, 104003, 104053, 104207, 104543, 104717, 104723, 105977, 106681, 108217, 108643, 108739, 110681, 110813, 111509, 111773, 111863, 112181, 112297, 112583, 112939, 113117, 113921, 114001, 114901, 115757, 115763, 119033, 119267, 119557, 119759, 120097, 120557, 120563, 120817, 120823, 121013, 121421, 121553, 122099, 122273, 122579, 122833, 123433, 124459, 125107, 125539, 125737, 126317, 126839, 127487, 127663, 127733, 128153, 128449, 128467, 128483, 129763, 130343, 130829, 131933.

Some larger examples :