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brilliant n.  and  Duffinian n.

First numbers that belong to both sets:

4, 9, 21, 25, 35, 49, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 713, 731, 737, 767, 779, 781, 793, 799, 803, 817, 841, 851, 869, 871, 893, 899, 901, 913, 923, 943, 949, 961, 979, 989, 1003, 1007, 1027, 1037, 1067, 1073, 1079, 1081, 1121, 1147, 1157, 1159, 1189, 1207, 1219, 1241, 1247, 1261, 1271, 1273, 1333, 1343, 1349, 1357, 1363, 1369, 1387, 1403, 1411, 1457, 1501, 1513, 1517, 1537, 1541, 1577, 1591, 1633, 1643, 1649, 1679, 1681, 1691, 1711, 1739, 1763, 1769, 1817, 1829, 1843, 1849, 1909, 1927, 1943, 1961, 2021, 2047, 2059, 2077, 2117, 2173, 2183, 2201, 2209, 2231, 2257, 2263, 2279, 2291, 2407, 2419, 2449, 2479, 2491, 2501, 2537, 2573, 2581, 2623, 2627, 2747, 2759, 2773, 2809, 2813, 2867, 2881, 2911, 2923, 2993, 3007, 3053, 3071, 3127, 3139, 3149, 3233, 3239, 3293, 3337, 3397, 3403, 3431, 3481, 3551, 3569, 3589, 3599, 3649, 3713, 3721, 3763, 3827, 3869, 3901, 3953, 3977, 4087, 4171, 4183, 4187, 4189, 4307, 4331, 4399, 4453, 4489, 4559, 4661, 4717, 4757, 4819, 4891, 4897, 5041, 5063, 5141, 5183, 5251, 5293, 5329, 5429, 5561, 5609, 5723, 5767, 5893, 5917, 5963, 6059, 6241, 6319, 6497, 6499, 6557, 6887, 6889, 7031, 7081, 7387, 7663, 7921, 8051, 8633, 9409, 10201, 10403, 10609, 10807, 11009, 11021, 11227, 11413, 11449, 11639, 11663, 11881, 12091, 12317, 12769, 12827, 13081, 13231, 13493, 13589, 13837, 13843, 14017, 14039, 14111, 14279, 14317, 14351, 14659, 14803, 14873, 14933, 15049, 15151, 15251, 15347, 15481, 15553, 15707, 15857, 15943, 16129, 16157, 16171, 16241, 16459, 16463, 16637.

Some larger examples :