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balanced n.  and  emirpimes

First numbers that belong to both sets:

15, 49, 169, 177, 178, 226, 302, 502, 551, 586, 589, 622, 781, 794, 817, 835, 899, 913, 961, 1202, 1346, 1366, 1454, 1766, 2206, 2219, 2234, 2402, 2426, 2443, 2446, 2458, 2474, 2603, 2605, 2614, 2642, 3099, 3117, 3155, 3173, 3207, 3365, 3401, 3459, 3473, 3521, 3603, 3622, 3651, 3653, 3665, 3715, 3721, 3777, 3778, 7201, 7223, 7251, 7343, 7367, 7445, 7715, 7751, 7799, 7837, 7849, 7913, 7957, 7967, 7971, 8135, 8159, 8309, 8321, 8345, 8351, 8399, 8507, 8542, 8557, 8566, 8593, 8735, 8759, 8797, 8903, 8909, 8935, 8945, 8983, 9014, 9038, 9069, 9259, 9263, 9326, 9335, 9383, 9411, 9427, 9458, 9487, 9501, 9502, 9518, 9529, 9543, 9578, 9611, 9641, 9659, 9663, 9755, 9771, 9799, 9813, 9818, 9827, 9897, 9938, 9953, 9983, 9987, 9995, 9997, 10003, 10006, 10021, 10033, 10057, 10081, 10117, 10129, 10229, 10231, 10265, 10279, 10319, 10327, 10347, 10349, 10358, 10411, 10454, 10474, 10519, 10523, 10541, 10553, 10637, 10643, 10661, 10673, 10694, 10699, 10705, 10713, 10721, 10747, 10749, 10798, 10805, 10838, 10893, 10954, 11015, 11029, 11051, 11091, 11141, 11146, 11195, 11237, 11267, 11293, 11303, 11309, 11318, 11335, 11341, 11363, 11366, 11378, 11413, 11453, 11459, 11465, 11479, 11531, 11539, 11558, 11569, 11651, 11653, 11654, 11714, 11723, 11729, 11769, 11787, 11797, 11841, 11843, 11857, 11881, 11963, 12169, 12233, 12237, 12335, 12367, 12398, 12403, 12407, 12439, 12469, 12494, 12531, 12559, 12571, 12587, 12602, 12605, 12629, 12634, 12643, 12679, 12683, 12686, 12707, 12709, 12746, 12769, 12851, 12871, 12877, 12891, 12913, 12949, 12977, 12997, 13106, 13138, 13193, 13201, 13273, 13387, 13418, 13453, 13459, 13507, 13511, 13521, 13541, 13543, 13561, 13579, 13582, 13589, 13637, 13641, 13657, 13666, 13703, 13705, 13714, 13843, 13891, 13897, 13922, 13985, 13991, 13993, 14005, 14017, 14059, 14111, 14117, 14129.

Some larger examples :