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powerful n.  and  self n.

First numbers that belong to both sets:

1, 9, 64, 108, 121, 288, 400, 512, 648, 1728, 2312, 3528, 5776, 6272, 6889, 7396, 7688, 8836, 9409, 9747, 10368, 10816, 10952, 11907, 12100, 12168, 12348, 13068, 17689, 18769, 19208, 21168, 22472, 22707, 23328, 25947, 27556, 27848, 29929, 30752, 30976, 33856, 34969, 35937, 37636, 42632, 44217, 45387, 46208, 47916, 49729, 50653, 54925, 59168, 64827, 65536, 69169, 69696, 70688, 70756, 75076, 75625, 76729, 77976, 78125, 80000, 80656, 86528, 87808, 98000, 110224, 111392, 123823, 124609, 126736, 128547, 129032, 132496, 133128, 134689, 136107, 137288, 146016, 154568, 156816, 162409, 165649, 177147, 190096, 195112, 196249, 202248, 206116, 208849, 212552, 213867, 215296, 223587, 249696, 254016, 264627, 270848, 275427, 275684, 276125, 279936, 284089, 286225, 287496, 288800, 298116, 300763, 310472, 314432, 320787, 332667, 344569, 356168, 369800, 373527, 382347, 388129, 389376, 390625, 390728, 396508, 401956, 405769, 414736, 419528, 427716, 452929, 458329, 463347, 465125, 467856, 480491, 484416, 497025, 508369, 509796, 512072, 512656, 518616, 521667, 524176, 524288, 525625, 528125, 528529, 561816, 566048, 582169, 599076, 600608, 615627, 617796, 629856, 645248, 646416, 657721, 664225, 665856, 677376, 681472, 695556, 710432, 715716, 734449, 744775, 746496, 751896, 766656, 767376, 771147, 780125, 799236, 804357, 814088, 822649, 845000, 853776, 861125, 861224, 868624, 876096, 884736, 889056, 893025, 898425, 937024, 940896, 968256, 970225, 978121, 1000000, 1000188, 1002528, 1013888, 1020100, 1067089, 1069227, 1075369, 1088856, 1089936, 1090827, 1114407, 1136356, 1137032, 1155625, 1175056, 1176125, 1177225, 1179396, 1181569, 1182816, 1199025, 1204352, 1218816, 1247616, 1250000, 1266325, 1275125, 1285956, 1288408, 1318492, 1336336, 1338248, 1352569, 1354896, 1357225, 1361367, 1377800, 1380625, 1382976, 1385329, 1401856, 1404488, 1440747, 1471369, 1481089, 1485172, 1498176, 1520768, 1527696, 1542267, 1548800, 1572516, 1591328, 1597536, 1605289, 1623076, 1648656, 1653125, 1661168, 1682000, 1692800, 1708249, 1722368, 1724976, 1726596, 1757600, 1758276, 1770125, 1803649, 1811187, 1819801, 1838736, 1841449, 1878268, 1881800, 1908576, 1909924, 1920996, 1921964, 1940449, 1946025, 1971216, 1990656, 2024072.

Some larger examples :