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powerful n.  and  Ulam n.

First numbers that belong to both sets:

1, 4, 8, 16, 36, 72, 243, 324, 400, 441, 800, 864, 1125, 1296, 1944, 2809, 2916, 3456, 3481, 3872, 4624, 4900, 5184, 6889, 7776, 8192, 10584, 11236, 11881, 13689, 14283, 15488, 17689, 18432, 19600, 19683, 20736, 21904, 24025, 25000, 25921, 29403, 34969, 36504, 39204, 45796, 47089, 47524, 47961, 56169, 59536, 65219, 72361, 73984, 75843, 78961, 84500, 84872, 87616, 88200, 95481, 97336, 99127, 102152, 106929, 114921, 120409, 123823, 127449, 141376, 147456, 154449, 160000, 163592, 172872, 175616, 177957, 184041, 190096, 196249, 197136, 197568, 203401, 210125, 224676, 242000, 253125, 253472, 272484, 274625, 292032, 293764, 295704, 303372, 304200, 311364, 313632, 323208, 338000, 342792, 346921, 349448, 350464, 352800, 356409, 376712, 379456, 383688, 385875, 386884, 390625, 391876, 392000, 393129, 400689, 405224, 405769, 412164, 419528, 451584, 455877, 460800, 469225, 484416, 484812, 493039, 501264, 504008, 521752, 527076, 553536, 574992, 575532, 585225, 592900, 609408, 616225, 620289, 625000, 634207, 663552, 665500, 672400, 680625, 682587, 685900, 692224, 702464, 704969, 708964, 712336, 765625, 779689, 781456, 788544, 793016, 864000, 874800, 884547, 912600, 940900, 950625, 964467, 975375, 984987, 985527, 996872, 1000000, 1010025, 1032256, 1058000, 1094116, 1104601, 1108809, 1114407, 1120581, 1147041, 1168561, 1182816, 1183744, 1194649, 1196836, 1203052, 1213488, 1218375, 1224963, 1279161, 1292832, 1299600, 1305728, 1318688, 1320201, 1364552, 1387684, 1397792, 1423656, 1444000, 1445000, 1454436, 1485172, 1488024, 1498176, 1512500, 1513800, 1516563, 1520768, 1527752, 1534752, 1540081, 1543500, 1547536, 1595169, 1602756, 1641672, 1672704, 1682209, 1687401, 1703025, 1711125, 1744712, 1752192, 1759688, 1769472, 1778112, 1812608, 1817104, 1830609, 1843200, 1893376, 1897352, 1898884, 1912689, 1934881, 1936000, 1946025, 1954404, 2025000, 2048288, 2105352, 2126696, 2131600, 2140369, 2154888, 2195200, 2196324, 2229049, 2247001, 2302128, 2328676, 2349675, 2413827, 2478175, 2481156, 2500000, 2505889, 2582449, 2596167, 2598544, 2598977, 2608328, 2621161, 2628288, 2640625, 2645163, 2653641, 2656192, 2661336, 2667168, 2669956, 2679769, 2695275, 2719201, 2725801, 2730348, 2768896, 2835856, 2856100, 2900209, 2941225, 2956824.

Some larger examples :