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brilliant n.  and  d-powerful n.

First numbers that belong to both sets:

4, 6, 9, 209, 377, 407, 629, 1073, 1349, 2263, 2407, 2449, 2537, 2623, 2627, 2773, 3569, 5329, 22663, 22733, 23587, 24289, 24823, 24883, 25273, 32437, 32743, 32899, 33221, 33227, 33823, 34387, 38383, 43363, 46843, 46973, 46979, 48863, 49729, 55483, 57263, 59249, 60409, 62357, 62429, 62449, 62519, 62609, 62663, 63293, 63433, 64643, 65473, 65693, 65783, 66049, 66887, 67259, 67693, 68753, 72389, 72833, 73537, 76529, 78563, 81253, 82249, 82427, 82937, 84757, 85273, 85279, 86153, 88489, 91829, 92303, 93421, 95023, 98437, 98503, 98723, 106439, 106499, 117983, 118397, 122783, 123223, 123353, 124237, 124307, 124387, 124723, 125263, 125977, 126389, 127759, 130799, 132427, 132467, 132577, 133729, 133927, 134429, 134557, 134773, 135767, 137297, 137549, 137741, 137749, 139259, 144293, 145247, 152227, 152687, 152731, 154393, 156517, 167539, 168437, 177329, 177439, 177923, 178463, 182293, 184363, 186293, 190427, 192397, 193279, 194393, 195589, 205273, 214769, 220397, 222359, 222559, 222973, 223943, 224173, 225763, 225893, 226243, 226489, 229367, 229673, 232949, 233627, 235247, 235757, 235759, 236237, 236273, 236941, 237667, 237803, 241763, 242939, 243697, 243983, 245743, 245813, 246269, 246377, 246973, 249323, 249349, 249479, 249493, 249761, 252347, 253087, 254563, 255373, 256433, 256453, 256457, 256583, 256637, 256769, 257267, 257333, 257573, 257843, 259447, 262793, 263147, 263563, 264203, 264373, 264421, 264427, 264623, 265219, 265723, 265927, 266243, 267859, 268483, 268867, 268951, 269459, 272543, 273223, 273883, 274613, 276491, 278857, 279229, 279287, 279379, 282223, 282533, 282593, 282883, 283963, 284629, 285143, 285347, 285653, 285671, 286283, 286597, 287213, 287543, 287659, 287879, 293627, 295739, 296411, 296893, 296897, 298273, 298783, 307469, 314843, 315619, 320857, 322363, 322697, 322853, 323429, 324563, 326699, 327937, 328231, 328459, 328583, 329467, 332429, 340943, 341689, 342439, 342523, 344459, 344569, 344657, 345269, 345283, 345623, 345641, 346211, 347573, 349823, 354049, 354379, 355253, 363287, 364729, 365221, 366257, 367153, 367643, 370229, 376973, 382297, 400289, 412679, 425639, 426373.

Some larger examples :