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amenable n.  and  frugal n.

First numbers that belong to both sets:

125, 128, 256, 512, 625, 729, 1024, 1029, 1280, 1369, 1536, 1681, 1701, 1792, 1849, 2048, 2197, 2209, 2401, 2560, 2809, 3125, 3481, 3584, 3645, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6889, 7921, 8192, 9409, 10000, 10112, 10125, 10201, 10240, 10368, 10449, 10496, 10609, 10624, 10625, 10633, 10752, 10976, 10985, 11008, 11045, 11125, 11392, 11421, 11449, 11664, 11776, 11881, 12005, 12032, 12125, 12288, 12393, 12416, 12500, 12544, 12769, 12800, 13125, 13568, 13824, 14045, 14336, 14337, 14641, 14749, 14848, 15104, 15309, 15552, 15616, 15625, 15872, 16000, 16121, 16129, 16281, 16384, 17152, 17161, 17253, 17405, 17496, 17920, 18125, 18176, 18225, 18605, 18688, 18769, 18944, 19197, 19208, 19321, 20000, 20169, 20224, 20237, 20480, 20577, 20736, 20992, 21141, 21248, 21609, 21952, 22016, 22201, 22445, 22784, 22801, 22981, 23125, 23328, 24057, 24064, 24353, 24389, 24565, 24576, 24649, 24832, 25000, 25088, 25205, 25625, 26244, 26569, 26645, 26973, 27097, 27136, 27889, 28125, 28469, 28561, 28672, 29889, 29929, 30208, 30625, 31104, 31205, 31213, 31232, 32000, 32041, 32761, 32768, 32805, 33125, 34304, 34445, 34992, 35721, 36352, 36481, 36501, 37249, 37376, 38125, 38416, 38637, 38809, 39601, 39605, 40000, 40448, 40625, 40817, 40960, 41472, 41553, 42496, 43904, 44469, 44521, 45568, 45625, 46656, 47045, 48013, 49152, 49664, 49729, 50000, 50301, 50421, 50625, 50653, 51529, 52441, 52488, 53125, 53217, 54289, 55625, 57121, 57344, 58081, 59049, 60025, 60625, 62208, 62500, 63001, 64000, 64881, 65536, 65625, 66049, 67228, 67797, 68921, 69169, 69629, 69984, 70713, 72361, 73205, 73441, 76545, 76729, 76832, 78125, 78732, 78961, 80000, 80089, 81920, 83349, 83521, 85169, 85293, 85849, 87808, 88837, 89373, 90625, 91125, 93312, 94041, 94249, 96721, 97969, 98304, 98441, 100000, 100352, 100489, 100625, 100736, 100845, 100864, 100949, 101120, 101125, 101185, 101632, 101817, 101888, 102016, 102400, 102541, 102625, 102656, 102789.

Some larger examples :