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balanced prime  and  cyclic n.

First numbers that belong to both sets:

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393, 5563, 5807, 6073, 6263, 6317, 6323, 6367, 6373, 6863, 6977, 7523, 7583, 7823, 7841, 8117, 8713, 8747, 9397, 9473, 9871, 10253, 10607, 10657, 10853, 11299, 11411, 11497, 11731, 11807, 11903, 11933, 12497, 12547, 12583, 12647, 12653, 12841, 12973, 13043, 13177, 13457, 13463, 14543, 14747, 14753, 15161, 15193, 15313, 15467, 15767, 15797, 15803, 15907, 15913, 16097, 16223, 16427, 16487, 16567, 16619, 16787, 16937, 16987, 17047, 17327, 17431, 17477, 17483, 17851, 18217, 18223, 18341, 18433, 18731, 19463, 19477, 19483, 19577, 19603, 19739, 19949, 20107, 20123, 20161, 20201, 20347, 20521, 20731, 21163, 21661, 21911, 21997, 22051, 22073, 22259, 22447, 23327, 23333, 23767, 23801, 23893, 24077, 24097, 24413, 24677, 25391, 25463, 25621, 25673, 26041, 26177, 26183, 26393, 26693, 26723, 26987, 27067, 27773, 28069, 28597, 28927, 29173, 29269, 29333, 29599, 29873, 30059, 30097, 30103, 30313, 30449, 30637, 30643, 30977, 31051, 31327, 32083, 32479, 32497, 32993, 33113, 33857, 34171, 34267, 34543, 34631, 34673, 34877, 35129, 35317, 35753, 35851, 35911, 36067, 36229, 36313, 36677, 36821, 37957, 38183, 38867, 39113, 39863, 40063, 40093, 40111, 40283, 40471, 40493, 40933, 40961, 41017, 41263, 41479, 41513, 41603, 41947, 42187, 42209, 42397, 42533, 42683, 43189, 44041, 44519, 44543, 44741, 45433, 45497, 45757, 46147, 46889, 47303, 47507, 47527, 47869, 48023, 48259, 48533, 48661, 49069, 49633, 50153, 50417, 50441, 51413, 51461, 51551, 52529, 52727, 52957, 53617, 53623, 53939, 54151, 54443, 54623, 54673, 54773, 54851, 55207, 55793, 55843, 56003, 56087, 56093, 56149, 56437, 56773, 56957, 57119, 57853, 58067, 58129, 58237.

Some larger examples :