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balanced n.  and  strong prime

First numbers that belong to both sets:

11, 37, 41, 149, 163, 197, 541, 557, 569, 587, 613, 617, 659, 787, 809, 929, 2141, 2203, 2281, 2333, 2347, 2381, 2389, 2467, 2473, 2617, 2707, 2729, 2851, 2857, 2897, 2953, 3461, 6791, 6857, 6947, 6959, 6991, 7013, 7039, 7151, 7187, 7243, 7477, 7487, 7499, 7517, 7559, 7877, 7949, 7993, 8087, 8147, 8161, 8219, 8287, 8311, 8363, 8429, 8443, 8521, 8537, 8563, 8597, 8627, 8641, 8677, 8689, 8819, 8861, 8969, 8999, 9011, 9059, 9157, 9173, 9277, 9281, 9319, 9337, 9371, 9419, 9511, 9547, 9613, 9619, 9767, 9781, 9803, 9811, 9829, 9923, 9929, 10177, 10267, 10271, 10301, 10331, 10357, 10391, 10453, 10457, 10597, 10781, 10949, 10987, 11027, 11057, 11113, 11239, 11251, 11317, 11351, 11369, 11437, 11443, 11467, 11489, 11587, 11593, 11617, 11657, 11699, 11801, 11831, 11969, 12037, 12041, 12071, 12097, 12157, 12161, 12343, 12391, 12409, 12451, 12473, 12487, 12601, 12697, 12713, 12739, 12889, 12907, 12941, 12953, 13001, 13093, 13217, 13327, 13397, 13411, 13417, 13441, 13451, 13513, 13591, 13721, 13859, 13873, 13901, 13997, 14029, 14051, 14071, 14369, 14387, 14401, 14407, 14419, 14447, 14533, 14561, 14621, 14627, 14699, 14737, 14779, 14867, 14887, 14923, 15013, 15121, 15137, 15359, 15373, 15427, 15493, 15619, 15641, 15817, 15859, 15877, 15887, 16001, 16061, 16127, 16139, 16229, 16301, 17021, 17093, 17539, 17623, 17827, 18311, 18397, 18793, 19207, 33149, 33199, 33343, 33391, 33403, 33487, 33529, 33679, 33703, 33739, 33749, 33911, 34127, 34141, 34157, 34253, 34259, 34351, 34421, 34483, 34537, 34583, 34589, 34603, 34649, 34757, 34939, 35023, 35051, 35251, 35419, 35437, 35507, 35527, 35531, 35569, 35747, 36007, 36011, 36209, 36293, 36451, 36493, 36629, 36739, 37039, 37181, 37199, 37337, 37361, 37483, 37547, 37561, 37589, 37781, 37991, 38039, 38069, 38167, 38197, 38219, 38609, 38669, 38693, 38723, 38971, 39019, 39079, 39097, 39191, 39239.

Some larger examples :