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odious n.  and  powerful n.

First numbers that belong to both sets:

1, 4, 8, 16, 25, 32, 49, 64, 81, 100, 121, 128, 196, 200, 256, 289, 324, 361, 392, 400, 484, 512, 529, 625, 648, 675, 784, 800, 841, 961, 968, 1024, 1089, 1125, 1156, 1225, 1296, 1444, 1521, 1568, 1600, 1681, 1849, 1936, 2048, 2116, 2187, 2197, 2312, 2401, 2500, 2592, 2601, 2700, 2888, 3025, 3136, 3200, 3364, 3481, 3844, 3872, 4096, 4225, 4232, 4356, 4489, 4500, 4563, 4624, 4900, 5000, 5041, 5184, 5400, 5625, 5776, 5929, 6084, 6241, 6272, 6400, 6724, 6728, 7225, 7396, 7569, 7688, 7744, 7921, 8192, 8281, 8464, 8575, 8712, 8748, 8788, 9000, 9025, 9248, 9409, 9604, 9800, 10000, 10201, 10368, 10404, 10609, 10800, 11552, 11907, 11979, 12100, 12167, 12168, 12544, 12769, 12800, 13448, 13456, 13689, 13924, 14641, 14792, 15376, 15488, 15625, 16129, 16384, 16641, 16807, 16900, 16928, 17161, 17424, 17496, 17576, 17956, 18000, 18225, 18252, 18496, 19208, 19321, 19600, 19773, 20000, 20164, 20449, 20736, 20808, 21025, 21600, 21609, 22201, 22500, 23104, 23409, 23716, 24200, 24336, 24389, 24649, 24964, 25088, 25600, 25947, 26569, 26896, 26912, 27848, 27889, 28125, 28561, 28900, 29584, 29929, 30276, 30625, 30752, 30976, 31684, 32041, 32761, 32768, 33124, 33275, 33800, 33856, 34225, 34300, 34848, 34992, 35152, 35721, 35912, 36000, 36100, 36504, 36992, 37249, 37636, 38416, 38809, 39200, 40000, 40328, 40401, 40804, 41067, 41472, 41616, 42436, 42875, 43200, 43681, 44217, 45000, 46208, 46225, 47089, 47432, 47628, 47916, 48400, 48668, 48672, 49729, 49928, 50176, 50625, 51076, 51200, 51529, 52441, 53361, 53792, 53824, 54675, 54756, 54925, 55225, 55696, 57121, 57800, 58564, 59049, 59168, 59643, 60552, 61009, 61504, 61952, 62500, 63001, 63368, 64516, 65219, 65536, 66049, 66248, 66564, 67081, 67228, 67600, 67712, 68600, 68644, 69169, 69696, 69984, 70227, 70304, 71289, 71824, 72000, 72200, 72900, 73008, 73441.

Some larger examples :