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balanced n.  and  good prime

First numbers that belong to both sets:

11, 37, 41, 149, 541, 557, 563, 569, 587, 809, 929, 2203, 2333, 3461, 6947, 7477, 7517, 8087, 8161, 8219, 8521, 8597, 8677, 8689, 8999, 9277, 9319, 9337, 9371, 9397, 9613, 10597, 10853, 11057, 11239, 11657, 11699, 12473, 12487, 12889, 14369, 14387, 14401, 14407, 14533, 14699, 15013, 17827, 19207, 35747, 36451, 38167, 38669, 39079, 40459, 40483, 41579, 41593, 43573, 43577, 44449, 46133, 47119, 47269, 47491, 47497, 50741, 51413, 52501, 52529, 54311, 54347, 54361, 55619, 56369, 61463, 61469, 64577, 132511, 132589, 134837, 134989, 136027, 136163, 136177, 136303, 144241, 148151, 148331, 148663, 148853, 149053, 150827, 150869, 151141, 151153, 156967, 157013, 159013, 159617, 160579, 165287, 165437, 166237, 168433, 168599, 168851, 168991, 169063, 169649, 175937, 176299, 176327, 177677, 178247, 178439, 178469, 178481, 181193, 181603, 181667, 184487, 184553, 185057, 185429, 190507, 190577, 201389, 201401, 201449, 201577, 204299, 205019, 205031, 205391, 206879, 206909, 207187, 207923, 209123, 218417, 219277, 227363, 228097, 234067, 234083, 234131, 234161, 234181, 236641, 236681, 237683, 237689, 237851, 237959, 237971, 238151, 243073, 246469, 246497, 250949, 254729, 535481, 540101, 541181, 547223, 547471, 547483, 562271, 567631, 568877, 568903, 568979, 569579, 569659, 575551, 580513, 580631, 584249, 584357, 584387, 584593, 584869, 592303, 592843, 600283, 601747, 601801, 603719, 603731, 610523, 610541, 611939, 615253, 615337, 628673, 631133, 631357, 631387, 631453, 639907, 643403, 643847, 644123, 658043, 659831, 669241, 669283, 669787, 669839, 669847, 669853, 677947, 678341, 678577, 679361, 683299, 683651, 683681, 686723, 686729, 690997, 691109, 691531, 691681, 701209, 701219, 701329, 703873, 704003, 707627, 707669, 730397, 730537, 730777, 747811, 748169, 750599, 750641, 750661, 750977, 757903, 757927, 757993, 758099, 758179, 762721, 763403, 764947, 764969, 775937, 780553, 789941, 789959, 790189, 791773, 791783, 791887, 803813, 804743, 808147, 808169, 809141, 809173, 821647, 822551, 838037, 840817, 846721, 846869, 846917, 854869, 854897, 855581, 856693, 859553, 861491, 861541, 863633, 872017.

Some larger examples :