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pernicious n.  and  perfect power

First numbers that belong to both sets:

9, 25, 36, 49, 81, 100, 121, 144, 196, 289, 324, 361, 400, 484, 529, 576, 625, 784, 841, 961, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1681, 1849, 1936, 2116, 2187, 2197, 2304, 2401, 2500, 2601, 3025, 3136, 3364, 3481, 3844, 4225, 4356, 4489, 4624, 4900, 5041, 5184, 5776, 5929, 6084, 6241, 6400, 6724, 7225, 7396, 7569, 7744, 8281, 8464, 9025, 9216, 9409, 9604, 10000, 10404, 10609, 12100, 12544, 12769, 13456, 13924, 14641, 15376, 15625, 16129, 16641, 16807, 16900, 17161, 17424, 17576, 17956, 18225, 18496, 19600, 20164, 20736, 21025, 21609, 23104, 23716, 24336, 24649, 24964, 25600, 26896, 28900, 29584, 30276, 30976, 32761, 33124, 33856, 34225, 35721, 36100, 36864, 37249, 37636, 38416, 40000, 41616, 42436, 42875, 43681, 46225, 47089, 48400, 49729, 50176, 50625, 51076, 51529, 53361, 53824, 55225, 55696, 58564, 61504, 62500, 64516, 66049, 66564, 67081, 67600, 68644, 69169, 69696, 71824, 72900, 73984, 77841, 78400, 80656, 82944, 84100, 86436, 88209, 91125, 92416, 94249, 94864, 95481, 97344, 97969, 98596, 99856, 101761, 102400, 107584, 115600, 118336, 121104, 121801, 123201, 123904, 125000, 127449, 131044, 131769, 132496, 132651, 135424, 136900, 139129, 140608, 140625, 142884, 143641, 144400, 146689, 147456, 148225, 148996, 150544, 153664, 160000, 160801, 164025, 165649, 166464, 169744, 174724, 177147, 178929, 180625, 184041, 184900, 188356, 193600, 198025, 198916, 200704, 201601, 202500, 204304, 205379, 206116, 208849, 213444, 215296, 218089, 219961, 220900, 222784, 223729, 226981, 229441, 234256, 235225, 237169, 243049, 246016, 250000, 251001, 258064, 263169, 264196, 265225, 266256, 268324, 269361, 270400, 274576, 276676, 278784, 279936, 287296, 290521, 291600, 294849, 295936, 297025, 299209, 300763, 311364, 312481, 313600, 319225, 321489, 322624, 326041, 330625, 331776, 336400, 339889, 343000, 345744, 352836, 354025, 357911, 369664, 371293, 375769, 376996, 379456, 381924, 383161, 389017, 389376, 391876, 394384, 399424, 405769, 407044, 409600, 423801, 430336.

Some larger examples :