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cubic n.  and  powerful n.

First numbers that belong to both sets:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112, 205379, 216000, 226981, 238328, 250047, 262144, 274625, 287496, 300763, 314432, 328509, 343000, 357911, 373248, 389017, 405224, 421875, 438976, 456533, 474552, 493039, 512000, 531441, 551368, 571787, 592704, 614125, 636056, 658503, 681472, 704969, 729000, 753571, 778688, 804357, 830584, 857375, 884736, 912673, 941192, 970299, 1000000, 1030301, 1061208, 1092727, 1124864, 1157625, 1191016, 1225043, 1259712, 1295029, 1331000, 1367631, 1404928, 1442897, 1481544, 1520875, 1560896, 1601613, 1643032, 1685159, 1728000, 1771561, 1815848, 1860867, 1906624, 1953125, 2000376, 2048383, 2097152, 2146689, 2197000, 2248091, 2299968, 2352637, 2406104, 2460375, 2515456, 2571353, 2628072, 2685619, 2744000, 2803221, 2863288, 2924207, 2985984, 3048625, 3112136, 3176523, 3241792, 3307949, 3375000, 3442951, 3511808, 3581577, 3652264, 3723875, 3796416, 3869893, 3944312, 4019679, 4096000, 4173281, 4251528, 4330747, 4410944, 4492125, 4574296, 4657463, 4741632, 4826809, 4913000, 5000211, 5088448, 5177717, 5268024, 5359375, 5451776, 5545233, 5639752, 5735339, 5832000, 5929741, 6028568, 6128487, 6229504, 6331625, 6434856, 6539203, 6644672, 6751269, 6859000, 6967871, 7077888, 7189057, 7301384, 7414875, 7529536, 7645373, 7762392, 7880599, 8000000, 8120601, 8242408, 8365427, 8489664, 8615125, 8741816, 8869743, 8998912, 9129329, 9261000, 9393931, 9528128, 9663597, 9800344, 9938375, 10077696, 10218313, 10360232, 10503459, 10648000, 10793861, 10941048, 11089567, 11239424, 11390625, 11543176, 11697083, 11852352, 12008989, 12167000, 12326391, 12487168, 12649337, 12812904, 12977875, 13144256, 13312053, 13481272, 13651919, 13824000, 13997521, 14172488, 14348907, 14526784, 14706125, 14886936, 15069223, 15252992, 15438249, 15625000, 15813251, 16003008, 16194277, 16387064, 16581375, 16777216, 16974593, 17173512, 17373979, 17576000, 17779581, 17984728, 18191447, 18399744, 18609625, 18821096, 19034163, 19248832, 19465109, 19683000, 19902511, 20123648, 20346417, 20570824, 20796875, 21024576, 21253933, 21484952, 21717639, 21952000.

Some larger examples :