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balanced prime  and  good prime

First numbers that belong to both sets:

5, 53, 257, 563, 593, 733, 1367, 2677, 5387, 6263, 7523, 9397, 10853, 11807, 28597, 30059, 30637, 40471, 41513, 41603, 41947, 51413, 52529, 55793, 56437, 73523, 84053, 107581, 107693, 108739, 115757, 115763, 120557, 120817, 120823, 132511, 144247, 157013, 159623, 169633, 185057, 194653, 207463, 216731, 217163, 227153, 240841, 243367, 263077, 267193, 295837, 300569, 302573, 311957, 315361, 324397, 339601, 370399, 371837, 389897, 430499, 463213, 463747, 492403, 524863, 545527, 562181, 584593, 611939, 644123, 657911, 658117, 659831, 683687, 683693, 691687, 691709, 701957, 702269, 703849, 703861, 718337, 745673, 748603, 780553, 780649, 821747, 838379, 863983, 885727, 923023, 958333, 960983, 1000099, 1000171, 1055839, 1055851, 1072763, 1074883, 1101097, 1146703, 1208657, 1210793, 1211489, 1237037, 1243343, 1243933, 1253723, 1256687, 1309531, 1338229, 1338331, 1349897, 1377281, 1418849, 1423237, 1425439, 1431113, 1444753, 1453877, 1471403, 1488749, 1534739, 1587563, 1602317, 1672301, 1691069, 1691093, 1737991, 1745693, 1763407, 1772467, 1820249, 1839353, 1842073, 1901699, 1906361, 1907567, 1976543, 2077577, 2086321, 2096687, 2103553, 2125457, 2141407, 2224879, 2262817, 2263433, 2325767, 2380333, 2385919, 2385953, 2413217, 2416003, 2440177, 2484491, 2518933, 2548537, 2566001, 2570189, 2608097, 2610583, 2613607, 2613707, 2629469, 2641357, 2644823, 2703593, 2737949, 2808307, 2879693, 2880149, 2884559, 2934467, 2961583, 3151487, 3179977, 3188399, 3196093, 3196163, 3253949, 3303149, 3308423, 3363629, 3364409, 3393317, 3393323, 3503249, 3527963, 3529943, 3579487, 3581651, 3763229, 3765233, 3771583, 3776569, 3797377, 3872767, 3877603, 3884989, 3885647, 3968807, 4021873, 4032773, 4032851, 4147873, 4155337, 4177741, 4181257, 4181717, 4240399, 4249423, 4289827, 4291979, 4299397, 4366633, 4371943, 4409269, 4410079, 4525811, 4538497, 4549639, 4549669, 4599941, 4602041, 4602077, 4753543, 4780379, 4780403, 4793501, 4794197, 4800641, 4858031, 4887587, 4942703, 5015083, 5022643, 5068909, 5116861, 5121191, 5130409, 5132117, 5236949, 5237209, 5315963, 5321263, 5327671, 5410283, 5436229, 5530981, 5534317, 5557037, 5605337, 5605667, 5605673, 5657027, 5669159, 5669617, 5681089, 5688461, 5689793, 5690261, 5771257, 5787751, 5973287, 5977253, 6004259, 6004283, 6050333, 6087833, 6097997, 6131303, 6193007, 6195737, 6249391, 6249791, 6253063, 6449731, 6474233, 6488089.

Some larger examples :