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Friedman n.  and  pernicious n.

First numbers that belong to both sets:

25, 121, 127, 289, 625, 1255, 1296, 1395, 1435, 1792, 2187, 2500, 2503, 2504, 2507, 2509, 2592, 3159, 3685, 4106, 4167, 4624, 5120, 5776, 6144, 6145, 9025, 9216, 10192, 10251, 10368, 10426, 10525, 10824, 11259, 11264, 11665, 12100, 12104, 12288, 12337, 12493, 12505, 12544, 12550, 12595, 12769, 12843, 12960, 13125, 13496, 13764, 13822, 13825, 13826, 13832, 14175, 14641, 15379, 15568, 15613, 15621, 15622, 15625, 15626, 15628, 15633, 15634, 15656, 15679, 15688, 16245, 16295, 16348, 16372, 16375, 16381, 16382, 16385, 16387, 16447, 16448, 16479, 16743, 16758, 16783, 16797, 16798, 16807, 17253, 17368, 17384, 17482, 17536, 17856, 17892, 17920, 17925, 18225, 18432, 18594, 19453, 19592, 19653, 19682, 19684, 19734, 19737, 19738, 19845, 20736, 21375, 21495, 21504, 21606, 21844, 21848, 21953, 22528, 23392, 23456, 23546, 23548, 24336, 24367, 24375, 24564, 24576, 24584, 24964, 25105, 25314, 25872, 26624, 26754, 26896, 28226, 28559, 28672, 28678, 28749, 28764, 28900, 29282, 29584, 29632, 31250, 31252, 31256, 32744, 32747, 32761, 32762, 32764, 32771, 32772, 32778, 32783, 32785, 32786, 32795, 32825, 32832, 32836, 32845, 32851, 32853, 32854, 33579, 33696, 34968, 34992, 35152, 35684, 35721, 35932, 35942, 36864, 37249, 37840, 38416, 38912, 39295, 39313, 39314, 39362, 39368, 39456, 41472, 42875, 43268, 43688, 44676, 45056, 45361, 45362, 45364, 45368, 45632, 45864, 45947, 46256, 46626, 46632, 46640, 46655, 46657, 46658, 46660, 46664, 46672, 46688, 47652, 48672, 48750, 48755, 48757, 48758, 49152, 49896, 51200, 51398, 52483, 52498, 53245, 53248, 53824, 54369, 54436, 55225, 57644, 59039, 59055, 59129, 59263, 59318, 59375, 59759, 62476, 62504, 62968, 63478, 63895, 63985, 64513, 65480, 65483, 65485, 65486, 65488, 65491, 65528, 65542, 65841, 67149, 67228, 67252, 67392, 68644, 68800, 69253, 69472, 69984, 69993, 73984, 74353, 74533, 76832, 76835, 78115, 78117, 78132, 78136, 78225, 78975, 81920, 82755, 82942, 82944, 83957.

Some larger examples :