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Duffinian n.  and  hoax n.

First numbers that belong to both sets:

85, 265, 319, 355, 361, 391, 517, 535, 913, 985, 1111, 1165, 1219, 1255, 1507, 1600, 1633, 1903, 1921, 2155, 2173, 2227, 2401, 2461, 2500, 2515, 2605, 2785, 2839, 2911, 2965, 3091, 3505, 3649, 3865, 3973, 4189, 4279, 4369, 4765, 4855, 4981, 5269, 5305, 5485, 5525, 5539, 5935, 6115, 6187, 6259, 6385, 6439, 6457, 6475, 6583, 6835, 7051, 7339, 7447, 7465, 7525, 7627, 7915, 8005, 8023, 8077, 8149, 8185, 8257, 8325, 8347, 8851, 9031, 9193, 9229, 9301, 9355, 9535, 9571, 9985, 10291, 10375, 10489, 10579, 10669, 10675, 10705, 11065, 11209, 11659, 11775, 11785, 11857, 11965, 12055, 12091, 12667, 12775, 12847, 12937, 12955, 13325, 13369, 13639, 13747, 13765, 14017, 14179, 14359, 14475, 14719, 14809, 14881, 15115, 15205, 15275, 15529, 15709, 15835, 15853, 15943, 16015, 16105, 16123, 16285, 16357, 16425, 16473, 16537, 16591, 16735, 17125, 17149, 17221, 17225, 17455, 17635, 17661, 17833, 17905, 18025, 18085, 18207, 18247, 18337, 18355, 18391, 18409, 18425, 18607, 18805, 18841, 18985, 19147, 19165, 19255, 19615, 20065, 20299, 20623, 20631, 20785, 20893, 20911, 21055, 21253, 21361, 21625, 21667, 21685, 21775, 21829, 21865, 21919, 21955, 22081, 22117, 22143, 22585, 22837, 22971, 23075, 23089, 23449, 23665, 23725, 23755, 23809, 23953, 24403, 24637, 24655, 25015, 25105, 25177, 25267, 25275, 25429, 25573, 25735, 25807, 26167, 26185, 26365, 26419, 26425, 26527, 26707, 26905, 27049, 27085, 27193, 27247, 27319, 27355, 27475, 27535, 27589, 28039, 28255, 28322, 28525, 28705, 28849, 29065, 29227, 29371, 29435, 29515, 29551, 29623, 29893, 30055, 30397, 30505, 30523, 30739, 30811, 30865, 30991, 31099, 31585, 31639, 31693, 31765, 32017, 32079, 32197, 32593, 32755, 32827, 32899, 32989, 33097, 33169, 33727, 33775, 33907, 34051, 34249, 34339, 34555, 34609, 34717, 34725, 34861, 34987, 35005, 35125, 35239, 35383, 35635, 36139, 36193, 36265, 36409, 36535, 37129, 37165.

Some larger examples :