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balanced prime  and  congruent n.

First numbers that belong to both sets:

5, 53, 157, 173, 257, 263, 373, 607, 653, 733, 1103, 1223, 1367, 1511, 2287, 2677, 2903, 3637, 3733, 4013, 4597, 5303, 5807, 6073, 6263, 6317, 6367, 6373, 6863, 7583, 7823, 8117, 9397, 9871, 10253, 10607, 10853, 11807, 11903, 11933, 12583, 12647, 12653, 12973, 13463, 14543, 15767, 15797, 16223, 16487, 16567, 17047, 17327, 17431, 17477, 18223, 18341, 19463, 19477, 19949, 20201, 21661, 21911, 21997, 22447, 23327, 23333, 23767, 23893, 24077, 24413, 24677, 25391, 25463, 25621, 26183, 26693, 27773, 28069, 28597, 28927, 29173, 29269, 29333, 29599, 30103, 30637, 30977, 31327, 32479, 33113, 34543, 34631, 34877, 35129, 35317, 35911, 36229, 36313, 36677, 36821, 37957, 38183, 39863, 40063, 40093, 40111, 40471, 40493, 40933, 40961, 41263, 41479, 42397, 42533, 43189, 44519, 44543, 44741, 45497, 45757, 47303, 47527, 47869, 48023, 48533, 48661, 49069, 51413, 51461, 51551, 52727, 52957, 53623, 54151, 54623, 54773, 55207, 56087, 56093, 56149, 56437, 56773, 56957, 57119, 57853, 58237, 58543, 58573, 58693, 60133, 60509, 61637, 62207, 62213, 62597, 62633, 63493, 63527, 63559, 63703, 64013, 65413, 65557, 65957, 66431, 67511, 67783, 68743, 69623, 72031, 72481, 72733, 73127, 73477, 74317, 74471, 74567, 75181, 75527, 75533, 76487, 77023, 77557, 77951, 78173, 78797, 79823, 79973, 80917, 81077, 81439, 81749, 82463, 82493, 82793, 83437, 84053, 84437, 85837, 86263, 86981, 87517, 89527, 89983, 90511, 90703, 91303, 92581, 93053, 95917, 96431, 97967, 98597, 99823, 101287, 103567, 103669, 104021, 104053, 104207, 104543, 104717, 105367, 105373, 105607, 105613, 107581, 107693, 108439, 108637, 110813, 110933, 111509, 111521, 111773, 111863, 112181, 112583, 113117, 114901, 115757, 116719, 116797, 117437, 117757, 118831, 119557, 119759, 120097, 120557, 120823, 121007, 121013, 121327, 121421, 121447, 123887, 126317, 126839, 127487, 127663, 127733, 130343, 130687, 130693, 130829, 131933, 132511, 132911, 132929, 133103, 133717, 134053, 134213, 134263, 134333, 134639, 135301, 135607.

Some larger examples :