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Duffinian n.  and  Friedman n.

First numbers that belong to both sets:

25, 121, 125, 128, 289, 343, 625, 1024, 1255, 1285, 1296, 2048, 2187, 2500, 2501, 2507, 2509, 2737, 2916, 3125, 3281, 3685, 4096, 4167, 4624, 5776, 6455, 8192, 9025, 9216, 9261, 10201, 10525, 11259, 11663, 11664, 11665, 11943, 12091, 12100, 12167, 12337, 12493, 12505, 12769, 12955, 13125, 13225, 13243, 13725, 13813, 13823, 14035, 14129, 14175, 14641, 14647, 15125, 15613, 15623, 15625, 15627, 15631, 15633, 15655, 15689, 15697, 16347, 16375, 16384, 16387, 16479, 16743, 16783, 16807, 17437, 18225, 18265, 19321, 19453, 19629, 19683, 19693, 20485, 20736, 21845, 21953, 22757, 24339, 24367, 24375, 24385, 24389, 24964, 25105, 26244, 26896, 26973, 27783, 27889, 28322, 28561, 28900, 29282, 29519, 29523, 29525, 29549, 29584, 29768, 29929, 30625, 31250, 31347, 32747, 32751, 32759, 32761, 32765, 32768, 32815, 32825, 32849, 32855, 32875, 32885, 33655, 34991, 34993, 36864, 37187, 37249, 37449, 38416, 39283, 41323, 41665, 42025, 43264, 43375, 43689, 43775, 44977, 45363, 45365, 45369, 45873, 45927, 45947, 45957, 46613, 46615, 46637, 46641, 46651, 46655, 46656, 46657, 46661, 46673, 46677, 46875, 52429, 52483, 52493, 53245, 53297, 53824, 54953, 55225, 58957, 58973, 59039, 59045, 59049, 59057, 59129, 59147, 59319, 59759, 61435, 62503, 63895, 63985, 65471, 65483, 65485, 65489, 65491, 65531, 65536, 67149, 68644, 69253, 69895, 69985, 70225, 73984, 74183, 75433, 76835, 77459, 78055, 78123, 78125, 78133, 78545, 78659, 82944, 83521, 83957, 85264, 87381, 91853, 93217, 97333, 97343, 98305, 98375, 98435, 100255, 100525, 101299, 102391, 102487, 102541, 102595, 102783, 103823, 104329, 104375, 104897, 104969, 104976, 104979, 104995, 105025, 105625, 106289, 107329, 107584, 108241, 109351, 109353, 109355, 109375, 112505, 112509, 112895, 112923, 114242, 114244, 114245, 114395, 114627, 114647, 114681, 114779, 114921, 115625, 115699, 116317, 116635, 116729, 116837, 117067, 117128, 117475, 117567, 117575, 117587, 117597, 117613, 117623, 117625, 117635, 117637, 117639, 117641, 117647.

Some larger examples :