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Duffinian n.  and  Smith n.

First numbers that belong to both sets:

4, 27, 85, 121, 265, 319, 355, 391, 517, 535, 576, 729, 913, 985, 1111, 1165, 1219, 1255, 1507, 1633, 1903, 1921, 2155, 2173, 2227, 2461, 2515, 2605, 2785, 2839, 2888, 2911, 2965, 3091, 3505, 3649, 3865, 3973, 4189, 4279, 4369, 4765, 4855, 4981, 5269, 5305, 5485, 5539, 5915, 5935, 6115, 6187, 6259, 6385, 6439, 6457, 6583, 6835, 7051, 7339, 7447, 7465, 7627, 7915, 8005, 8023, 8077, 8149, 8185, 8257, 8347, 8851, 9031, 9193, 9229, 9301, 9355, 9535, 9571, 9985, 10201, 10291, 10489, 10579, 10669, 10705, 11065, 11209, 11659, 11785, 11857, 11965, 12055, 12091, 12667, 12847, 12937, 12955, 13369, 13639, 13747, 13765, 14017, 14179, 14359, 14719, 14809, 14881, 15115, 15205, 15529, 15709, 15835, 15853, 15943, 16015, 16105, 16123, 16285, 16357, 16537, 16591, 16735, 17149, 17221, 17424, 17455, 17635, 17833, 17885, 17905, 18085, 18247, 18337, 18355, 18391, 18409, 18496, 18607, 18805, 18841, 18985, 19147, 19165, 19255, 19615, 19683, 19943, 20065, 20299, 20623, 20785, 20893, 20911, 21055, 21253, 21361, 21667, 21685, 21829, 21865, 21919, 21955, 22081, 22117, 22585, 22837, 22971, 23089, 23449, 23665, 23755, 23809, 23953, 24403, 24637, 24655, 25015, 25105, 25177, 25267, 25429, 25573, 25735, 25807, 26167, 26185, 26365, 26419, 26527, 26707, 26905, 27049, 27085, 27193, 27209, 27247, 27319, 27355, 27535, 27589, 28039, 28255, 28705, 28849, 29065, 29227, 29371, 29515, 29551, 29623, 29893, 30055, 30397, 30505, 30523, 30739, 30811, 30865, 30991, 31099, 31585, 31639, 31693, 31765, 32017, 32197, 32593, 32755, 32827, 32899, 32989, 33097, 33169, 33635, 33727, 33907, 34051, 34249, 34339, 34555, 34609, 34717, 34861, 34987, 35005, 35239, 35383, 35635, 36139, 36193, 36265, 36335, 36409, 36481, 36535, 37129, 37165, 37237, 37255, 37615, 37651, 37705, 37885, 37903, 38029, 38137, 38263, 38515, 38641, 38785, 38911, 38965, 39073, 39127, 39253, 39271, 39307, 39347, 39505.

Some larger examples :