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powerful n.  and  unprimeable n.

First numbers that belong to both sets:

200, 324, 512, 625, 1352, 1764, 2048, 6400, 6724, 6728, 7776, 9216, 10125, 10404, 10648, 10800, 10816, 11664, 12996, 13500, 14112, 15488, 16200, 17424, 17672, 18000, 19044, 19652, 20000, 21296, 21632, 21904, 21952, 22472, 23104, 24300, 25000, 26136, 26912, 27000, 27225, 30375, 30625, 31104, 32768, 33124, 33275, 34225, 34992, 35344, 36125, 36504, 36864, 38025, 38416, 38988, 40328, 42592, 42632, 42875, 43904, 44944, 45796, 46208, 46656, 48600, 49284, 50000, 51984, 52272, 53824, 54925, 56644, 60516, 60552, 61952, 63504, 64800, 66125, 66248, 67228, 67500, 67712, 68600, 68644, 69984, 72000, 72200, 73984, 74088, 76176, 77284, 79092, 79524, 80000, 81608, 82944, 84100, 85184, 85264, 86400, 86436, 86528, 87025, 88200, 89888, 91204, 93025, 94864, 95048, 95832, 96100, 96800, 97200, 97336, 97344, 98784, 101124, 102152, 105125, 105625, 105800, 111132, 111392, 111556, 112896, 114244, 118336, 119072, 119164, 119716, 121104, 123904, 124416, 127008, 128000, 128164, 132300, 133225, 133956, 134456, 136125, 136900, 137288, 142884, 143648, 144000, 144500, 145924, 147968, 150544, 152100, 152352, 153125, 153664, 154568, 155952, 156025, 156800, 157464, 159048, 161312, 161604, 163592, 166375, 166464, 168200, 169744, 170528, 171125, 171396, 171500, 175232, 175616, 176868, 178084, 181476, 181656, 182408, 183184, 189728, 190512, 191844, 192200, 194400, 195112, 195364, 197192, 198025, 198916, 199712, 200704, 202500, 206116, 207576, 211600, 216000, 216225, 217800, 218700, 219024, 219488, 220500, 221184, 223112, 225000, 225792, 228484, 228488, 230400, 231125, 231200, 231868, 232324, 235224, 235225, 236196, 236672, 243000, 245000, 247808, 248004, 249696, 250632, 253125, 253472, 254016, 256000, 256328, 260100, 262088, 264196, 264500, 265225, 266200, 266256, 268324, 272484, 273375, 275684, 276125, 278784, 279936, 281216, 282752, 286225, 289444, 291600, 291848, 297675, 298116, 300125, 301088, 304175, 304200, 307328, 310472, 311364, 311904, 314432, 315844, 318096, 320356, 324000, 325125, 327184, 328536, 332024, 334084, 336200, 337500, 338000, 338724, 339488, 340736, 342225, 343000, 348100, 349448, 350464.

Some larger examples :