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Friedman n.  and  odious n.

First numbers that belong to both sets:

25, 121, 127, 128, 289, 625, 1022, 1024, 1255, 1296, 1395, 1435, 1503, 1792, 2048, 2187, 2500, 2503, 2504, 2507, 2509, 2592, 3159, 3685, 4088, 4096, 4106, 4167, 4624, 5776, 6145, 7928, 8092, 8192, 9025, 10192, 10201, 10251, 10368, 10426, 10525, 10824, 10935, 11163, 11259, 11264, 11439, 11663, 11665, 11943, 12006, 12100, 12103, 12104, 12107, 12109, 12167, 12337, 12493, 12505, 12544, 12550, 12595, 12769, 12843, 12960, 13125, 13286, 13496, 13689, 13725, 13764, 13822, 13825, 13826, 13832, 14035, 14175, 14641, 14647, 15003, 15030, 15246, 15379, 15568, 15613, 15621, 15622, 15625, 15626, 15628, 15631, 15633, 15634, 15645, 15655, 15656, 15661, 15662, 15667, 15679, 15688, 16225, 16245, 16295, 16348, 16372, 16375, 16381, 16382, 16384, 16387, 16447, 16479, 16743, 16758, 16783, 16797, 16798, 16807, 16879, 17253, 17368, 17384, 17482, 17536, 17856, 17892, 17920, 17925, 18225, 18270, 18594, 19321, 19453, 19592, 19653, 19682, 19684, 19693, 19734, 19737, 19738, 19773, 19845, 20736, 21175, 21375, 21495, 21504, 21606, 21753, 21844, 21848, 21870, 21875, 21953, 22264, 22528, 23326, 23392, 23456, 23546, 23548, 23796, 24336, 24339, 24367, 24375, 24389, 24390, 24393, 24564, 24584, 24964, 25105, 25314, 25375, 25725, 25872, 26348, 26624, 26754, 26896, 26973, 27889, 28217, 28226, 28547, 28554, 28556, 28559, 28561, 28564, 28672, 28678, 28749, 28764, 28900, 29282, 29517, 29523, 29525, 29526, 29529, 29584, 29632, 29929, 30625, 31250, 31252, 31256, 31346, 31509, 31682, 32744, 32747, 32761, 32762, 32764, 32768, 32771, 32778, 32783, 32785, 32786, 32795, 32825, 32836, 32845, 32851, 32853, 32854, 33579, 33655, 33696, 34968, 34992, 35152, 35684, 35721, 35932, 35942, 37249, 37814, 37840, 37845, 38416, 38912, 39283, 39295, 39313, 39314, 39342, 39356, 39362, 39368, 39382, 39456, 39864, 41468, 41472, 42875, 43268, 43688, 43691, 43932, 44676, 45056, 45361, 45362, 45364, 45367, 45368, 45632, 45864, 45947, 45978, 46256, 46615.

Some larger examples :