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cubic n.  and  gapful n.

First numbers that belong to both sets:

1000, 1331, 1728, 8000, 24389, 27000, 39304, 110592, 125000, 175616, 216000, 250047, 274625, 314432, 421875, 512000, 592704, 970299, 1000000, 1061208, 1157625, 1259712, 1331000, 1560896, 1601613, 1728000, 1771561, 2197000, 2460375, 2744000, 2985984, 3241792, 3375000, 3511808, 3796416, 4096000, 4492125, 4741632, 4913000, 5268024, 5832000, 6229504, 7414875, 7762392, 8000000, 9261000, 10077696, 10648000, 10941048, 11239424, 11390625, 11852352, 12167000, 12326391, 13144256, 13824000, 15069223, 15625000, 16003008, 16581375, 16777216, 17173512, 17576000, 19683000, 20346417, 20796875, 21952000, 23149125, 23393656, 24389000, 25412184, 25672375, 27000000, 27818127, 28372625, 29218112, 29791000, 31255875, 35287552, 35937000, 36926037, 37933056, 39651821, 41063625, 42875000, 46656000, 50653000, 54872000, 56623104, 57066625, 59319000, 62099136, 65939264, 74088000, 85184000, 85766121, 91125000, 94818816, 98611128, 100544625, 102503232, 103823000, 107850176, 110592000, 117649000, 119095488, 119823157, 121287375, 122023936, 123505992, 125000000, 128024064, 130323843, 132651000, 137388096, 140608000, 142236648, 144703125, 147197952, 148877000, 153990656, 157464000, 166375000, 168196608, 170953875, 171879616, 173741112, 175616000, 176558481, 185193000, 189119224, 191102976, 195112000, 197137368, 200201625, 205379000, 209584584, 210644875, 216000000, 218167208, 219256227, 221445125, 225866529, 226981000, 232608375, 238328000, 242970624, 244140625, 250047000, 253636137, 256047875, 259694072, 262144000, 268336125, 274625000, 275894451, 279726264, 281011375, 287496000, 291434247, 294079625, 296740963, 325660672, 328509000, 337153536, 347428927, 354894912, 363994344, 373248000, 376367048, 382657176, 385828352, 397065375, 405224000, 420189749, 421875000, 438976000, 447697125, 456533000, 474552000, 493039000, 496793088, 512000000, 519718464, 531441000, 535387328, 551368000, 561515625, 563559976, 571787000, 580093704, 592704000, 601211584, 603351125, 644972544, 653972032, 658503000, 690807104, 700227072, 753571000, 766060875, 773620632, 817400375, 830584000, 846590536, 884736000, 970299000, 988047936, 1000000000, 1003003001, 1006012008, 1009027027, 1015075125, 1024192512, 1030301000, 1048772096, 1061208000, 1083206683, 1092727000, 1099104768, 1108717875, 1111934656, 1118386872, 1124864000, 1157625000, 1177583616, 1191016000, 1197770328, 1204550144, 1207949625, 1218186432, 1225043000, 1245766976, 1259712000, 1295029000, 1302170688, 1312932375, 1316532736, 1323753192, 1331000000, 1367631000, 1371330631, 1389928896, 1404928000, 1412467848, 1423828125, 1435249152, 1442897000, 1458274104, 1466003456, 1481544000, 1520875000, 1528823808, 1540798875, 1544804416, 1548816893, 1552836312, 1556862679, 1560896000, 1601613000, 1613964717, 1626379776, 1643032000, 1651400568, 1664006625, 1676676672, 1685159000, 1710777536, 1728000000, 1745337664, 1771561000, 1780360128, 1793613375, 1798045696, 1806932232, 1815848000, 1820316861, 1860867000, 1888232256, 1906624000.

Some larger examples :