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amenable n.  and  Friedman n.

First numbers that belong to both sets:

25, 121, 125, 128, 153, 216, 289, 625, 688, 736, 1024, 1260, 1285, 1296, 1792, 2048, 2349, 2500, 2501, 2504, 2505, 2508, 2509, 2592, 2737, 2916, 3125, 3281, 3685, 3784, 3864, 3972, 4088, 4096, 4536, 4624, 4628, 5120, 5776, 5832, 6144, 6145, 6880, 7928, 8092, 8192, 9025, 9216, 9261, 10192, 10201, 10368, 10521, 10525, 10824, 11025, 11264, 11664, 11665, 11844, 11848, 12060, 12100, 12101, 12104, 12105, 12108, 12109, 12288, 12321, 12337, 12384, 12493, 12505, 12544, 12600, 12768, 12769, 12960, 12964, 12996, 13125, 13176, 13225, 13496, 13545, 13689, 13725, 13764, 13813, 13824, 13825, 13832, 14129, 14168, 14256, 14352, 14641, 14645, 15125, 15300, 15345, 15552, 15568, 15585, 15612, 15613, 15617, 15620, 15621, 15624, 15625, 15628, 15629, 15632, 15633, 15641, 15645, 15656, 15661, 15688, 15689, 15697, 16225, 16245, 16272, 16348, 16368, 16372, 16377, 16381, 16384, 16385, 16448, 16765, 16797, 17253, 17325, 17328, 17368, 17384, 17428, 17437, 17488, 17536, 17689, 17856, 17892, 17920, 17925, 18225, 18265, 18432, 18744, 19321, 19392, 19453, 19592, 19629, 19653, 19684, 19692, 19693, 19732, 19736, 19737, 19773, 19845, 20485, 20736, 21504, 21753, 21844, 21845, 21848, 21952, 21953, 22264, 22528, 22757, 23328, 23392, 23456, 23548, 23552, 23796, 24336, 24385, 24389, 24393, 24564, 24576, 24584, 24768, 24964, 24972, 25105, 25137, 25725, 25872, 25921, 26244, 26348, 26364, 26496, 26624, 26896, 26973, 27436, 27648, 27653, 27889, 28217, 28224, 28476, 28556, 28561, 28564, 28672, 28728, 28749, 28764, 28784, 28900, 29160, 29184, 29517, 29524, 29525, 29529, 29549, 29584, 29632, 29768, 29929, 30625, 31252, 31256, 31509, 32685, 32697, 32744, 32761, 32764, 32765, 32768, 32772, 32781, 32785, 32805, 32825, 32832, 32836, 32845, 32849, 32853, 32865, 32885, 33696, 34425, 34968, 34992, 34993, 34996, 35152, 35684, 35721, 35928, 35932, 35933, 35937, 36457, 36549, 36864, 37249, 37449, 37668, 37840, 37845, 38416.

Some larger examples :