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d-powerful n.  and  nude n.

First numbers that belong to both sets:

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 132, 135, 175, 224, 264, 333, 1326, 2136, 2224, 2244, 2248, 2664, 2824, 3324, 3432, 3492, 3612, 4224, 4244, 4248, 4288, 4332, 4424, 4644, 4932, 8224, 8288, 9324, 12324, 12432, 13332, 17248, 18872, 22332, 22848, 22932, 23232, 23292, 23364, 23436, 24264, 24336, 24624, 24848, 24864, 26244, 26448, 29232, 32364, 32436, 32832, 32922, 33222, 33264, 33288, 33624, 33648, 33864, 34236, 34272, 36324, 39393, 42244, 42248, 42264, 42624, 42864, 43212, 43236, 46824, 48264, 49896, 49968, 62664, 62688, 63324, 63432, 64224, 64332, 64464, 82248, 82448, 84224, 117726, 122436, 124632, 131232, 131292, 131322, 131328, 131364, 131412, 131832, 132264, 132444, 132624, 132648, 133212, 133326, 133344, 133728, 134736, 136464, 164832, 172662, 177324, 183624, 184744, 192312, 212364, 212436, 213264, 214236, 222224, 223236, 223272, 223632, 224316, 224664, 226332, 226464, 229968, 232236, 232368, 233244, 233262, 236124, 236232, 236328, 236364, 236436, 243324, 243432, 243636, 243768, 244332, 244776, 246336, 246372, 246624, 248664, 262224, 262248, 262332, 262464, 262488, 262644, 262662, 262824, 263124, 263184, 263232, 263322, 263328, 263364, 263412, 263436, 263832, 264264, 264288, 264312, 264336, 264444, 264624, 264648, 266262, 266448, 268248, 268824, 268992, 272832, 274344, 281232, 282264, 282336, 282624, 282648, 282824, 282848, 282864, 283368, 283632, 284224, 284288, 284688, 286224, 286464, 286824, 288264, 288648, 296496, 297864, 311724, 312444, 312864, 316644, 321222, 321264, 322164, 322236, 322326, 322362, 322368, 322632, 323226, 323244, 323424, 323622, 323664, 324324, 324612, 324636, 326322, 326328, 326412, 326616, 326676, 326832, 328272, 328416, 328632, 332262, 332622, 333126, 333432, 334248, 334332, 334464, 334488, 336222, 336264, 336444, 339696, 339966, 341244, 342324, 342636, 343644, 344124, 344364, 346164, 346212, 346236, 346368, 347424, 348696, 362232, 362364, 362832, 363222, 363312, 363624, 364236, 364476, 366432, 366696, 367332, 368328, 369396, 369936, 371364, 372624, 373632, 386832, 399366, 416472, 423636, 427896, 428224, 432768.

Some larger examples :