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frugal n.  and  perfect power

First numbers that belong to both sets:

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192, 9409, 10000, 10201, 10609, 11449, 11664, 11881, 12167, 12544, 12769, 13824, 14641, 15625, 16129, 16384, 16807, 17161, 18225, 18769, 19321, 19683, 20736, 21609, 21952, 22201, 22801, 24389, 24649, 26244, 26569, 27889, 28561, 29791, 29929, 30625, 32041, 32761, 32768, 35721, 36481, 37249, 38416, 38809, 39601, 40000, 42875, 44521, 46656, 49729, 50625, 50653, 51529, 52441, 54289, 57121, 58081, 59049, 60025, 62500, 63001, 64000, 65536, 66049, 68921, 69169, 72361, 73441, 76729, 78125, 78961, 79507, 80089, 83521, 85849, 91125, 94249, 96721, 97969, 100000, 100489, 102400, 103823, 104976, 105625, 107584, 108241, 109561, 110224, 110592, 110889, 112225, 113569, 114244, 117649, 118336, 120409, 121801, 123201, 124609, 125000, 126025, 126736, 128881, 130321, 131072, 131769, 132651, 133225, 134689, 135424, 136161, 137641, 139129, 140608, 140625, 141376, 143641, 146689, 147456, 148877, 149769, 150544, 151321, 153664, 156025, 157464, 157609, 160000, 160801, 161051, 164025, 166375, 167281, 170569, 172225, 175561, 175616, 177147, 177241, 178929, 179776, 180625, 182329, 185193, 185761, 186624, 187489, 192721, 194481, 195112, 196249, 198025, 200704, 201601, 205379, 208849, 210681, 212521, 214369, 215296, 218089, 219961, 222784, 225625, 226981, 227529, 229441, 234256, 235225, 236196, 237169, 238144, 238328, 241081, 246016, 247009, 248832, 249001, 250000, 250047, 253009, 257049, 259081, 261121, 262144, 263169, 271441, 273529, 274625, 279841, 279936, 281961, 287296, 290521, 292681, 299209, 300763, 301401, 305809, 310249, 314432, 316969, 321489, 322624, 323761, 326041, 328509, 330625, 331776, 332929, 334084, 337561, 341056, 344569, 350464, 351649, 357911, 358801, 361201, 363609, 366025, 368449, 371293, 373248, 375769, 380689, 383161, 385641, 388129, 389017, 390625, 398161, 399424, 405224, 405769, 408321, 409600, 410881, 413449, 418609, 419904, 421875, 426409, 430336, 431649, 434281, 436921.

Some larger examples :