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odious n.  and  perfect power

First numbers that belong to both sets:

1, 4, 8, 16, 25, 32, 49, 64, 81, 100, 121, 128, 196, 256, 289, 324, 361, 400, 484, 512, 529, 625, 784, 841, 961, 1024, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1681, 1849, 1936, 2048, 2116, 2187, 2197, 2401, 2500, 2601, 3025, 3136, 3364, 3481, 3844, 4096, 4225, 4356, 4489, 4624, 4900, 5041, 5184, 5625, 5776, 5929, 6084, 6241, 6400, 6724, 7225, 7396, 7569, 7744, 7921, 8192, 8281, 8464, 9025, 9409, 9604, 10000, 10201, 10404, 10609, 12100, 12167, 12544, 12769, 13456, 13689, 13924, 14641, 15376, 15625, 16129, 16384, 16641, 16807, 16900, 17161, 17424, 17576, 17956, 18225, 18496, 19321, 19600, 20164, 20449, 20736, 21025, 21609, 22201, 22500, 23104, 23409, 23716, 24336, 24389, 24649, 24964, 25600, 26569, 26896, 27889, 28561, 28900, 29584, 29929, 30276, 30625, 30976, 31684, 32041, 32761, 32768, 33124, 33856, 34225, 35721, 36100, 37249, 37636, 38416, 38809, 40000, 40401, 40804, 41616, 42436, 42875, 43681, 46225, 47089, 48400, 49729, 50176, 50625, 51076, 51529, 52441, 53361, 53824, 54756, 55225, 55696, 57121, 58564, 59049, 61009, 61504, 62500, 63001, 64516, 65536, 66049, 66564, 67081, 67600, 68644, 69169, 69696, 71289, 71824, 72900, 73441, 73984, 75625, 77284, 77841, 78400, 79507, 80089, 80656, 81225, 81796, 82944, 84100, 86436, 88209, 88804, 90000, 90601, 91125, 92416, 93025, 93636, 94249, 94864, 95481, 97336, 97344, 97969, 98596, 99856, 101761, 102400, 104329, 105625, 106276, 107584, 108241, 111556, 112225, 114244, 115600, 116281, 117649, 118336, 119025, 119716, 121104, 121801, 122500, 123201, 123904, 124609, 125000, 126025, 126736, 127449, 128164, 131044, 131072, 131769, 132496, 132651, 135424, 136900, 139129, 140608, 140625, 142884, 143641, 144400, 146689, 148225, 148996, 150544, 151321, 153664, 154449, 155236, 156025, 160000, 160801, 161604, 163216, 164025, 165649, 166464, 167281, 169744, 174724, 177147, 177241, 178929, 180625, 184041, 184900, 185761, 187489, 188356, 189225, 192721, 193600.

Some larger examples :