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Duffinian n.  and  Ulam n.

First numbers that belong to both sets:

4, 8, 16, 36, 57, 77, 155, 175, 189, 209, 219, 221, 243, 253, 309, 319, 324, 341, 363, 400, 451, 485, 497, 605, 685, 722, 781, 800, 849, 893, 905, 927, 949, 1025, 1037, 1079, 1081, 1101, 1157, 1191, 1296, 1313, 1355, 1387, 1389, 1465, 1641, 1643, 1682, 1765, 1853, 1919, 1985, 2117, 2249, 2327, 2511, 2513, 2525, 2581, 2613, 2635, 2721, 2723, 2809, 2811, 2831, 2899, 2916, 2921, 2987, 3043, 3065, 3085, 3107, 3131, 3153, 3205, 3207, 3239, 3261, 3263, 3305, 3415, 3451, 3473, 3481, 3483, 3591, 3605, 3649, 3698, 3872, 3991, 4121, 4165, 4167, 4187, 4341, 4343, 4453, 4531, 4619, 4624, 4707, 4881, 4883, 4925, 4927, 4971, 5079, 5159, 5184, 5269, 5291, 5335, 5357, 5489, 5533, 5597, 5599, 5663, 5665, 5687, 5765, 5797, 5853, 5875, 6049, 6051, 6107, 6239, 6283, 6325, 6393, 6437, 6457, 6479, 6635, 6721, 6865, 6889, 6931, 7041, 7063, 7217, 7275, 7427, 7461, 7471, 7693, 7779, 7781, 7977, 7979, 8057, 8077, 8143, 8165, 8192, 8255, 8275, 8451, 8453, 8509, 8531, 8553, 8639, 8705, 8749, 8751, 8771, 8773, 8815, 8873, 8917, 8947, 8949, 8983, 9071, 9179, 9193, 9313, 9355, 9379, 9399, 9401, 9443, 9509, 9553, 9575, 9577, 9621, 9641, 9665, 9853, 9917, 9939, 9961, 9983, 9985, 10085, 10107, 10173, 10249, 10269, 10349, 10393, 10415, 10523, 10525, 10537, 10547, 10569, 10655, 10677, 10679, 10713, 10757, 10865, 10911, 10931, 10997, 10999, 11099, 11107, 11217, 11219, 11236, 11253, 11275, 11603, 11605, 11627, 11647, 11713, 11735, 11737, 11757, 11759, 11881, 12055, 12145, 12187, 12365, 12443, 12482, 12707, 12749, 12751, 12785, 12873, 12961, 13069, 13071, 13157, 13181, 13303, 13369, 13401, 13531, 13565, 13609, 13611, 13655, 13675, 13733, 13917, 13919, 13953, 13975, 14129, 14195, 14283, 14305, 14349, 14427, 14491, 14493, 14579.

Some larger examples :