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cyclic n.  and  Smith n.

First numbers that belong to both sets:

85, 265, 319, 391, 517, 535, 895, 913, 985, 1111, 1165, 1219, 1507, 1633, 1795, 1903, 1921, 2173, 2227, 2461, 2515, 2785, 2839, 2911, 2965, 3091, 3595, 3649, 3865, 3973, 4189, 4279, 4369, 4765, 4981, 5071, 5269, 5485, 5539, 5935, 6115, 6187, 6259, 6295, 6385, 6439, 6457, 6583, 6835, 7051, 7195, 7249, 7339, 7447, 7465, 7627, 7915, 8023, 8077, 8095, 8149, 8185, 8257, 8347, 8545, 8653, 8851, 9031, 9193, 9229, 9301, 9427, 9535, 9571, 9895, 9985, 10291, 10489, 10579, 10669, 11065, 11209, 11659, 11695, 11785, 11857, 11965, 12091, 12667, 12847, 12937, 13369, 13639, 13747, 13765, 14017, 14179, 14359, 14719, 14809, 14881, 15115, 15529, 15709, 15835, 15853, 15943, 16015, 16123, 16269, 16285, 16357, 16537, 16591, 16645, 16735, 17149, 17221, 17635, 17833, 18085, 18247, 18337, 18391, 18409, 18607, 18841, 18895, 18985, 19147, 19165, 19615, 20065, 20299, 20623, 20695, 20785, 20893, 20911, 21253, 21361, 21667, 21685, 21829, 21865, 21919, 22081, 22117, 22585, 22837, 23089, 23449, 23665, 23809, 23953, 24259, 24403, 24637, 25015, 25177, 25195, 25267, 25429, 25573, 25735, 25807, 26167, 26185, 26365, 26367, 26419, 26527, 26545, 26707, 26995, 27049, 27085, 27193, 27247, 27285, 27319, 27535, 27589, 28039, 28635, 28849, 29065, 29227, 29371, 29461, 29515, 29551, 29623, 29695, 29893, 30397, 30523, 30739, 30811, 30865, 30991, 31099, 31585, 31639, 31693, 31765, 32017, 32593, 32827, 32899, 32989, 33097, 33169, 33727, 33907, 34051, 34249, 34339, 34609, 34663, 34717, 34861, 34987, 35095, 35239, 35293, 35383, 35545, 35635, 36139, 36193, 36265, 36409, 36535, 37129, 37165, 37237, 37615, 37651, 37795, 37885, 37903, 38029, 38137, 38263, 38515, 38641, 38785, 38911, 38965, 39073, 39127, 39253, 39271, 39307, 39523, 39595, 39685, 39757, 39811, 40567, 40585, 40711, 40981, 41089, 41359, 41557, 41917, 42493, 42565, 42691, 42817, 43357, 43429, 43501, 43699, 43807, 43861, 43879, 44167.

Some larger examples :