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balanced n.  and  Friedman n.

First numbers that belong to both sets:

153, 216, 625, 1022, 2349, 2501, 2502, 2505, 2506, 2508, 2737, 2916, 3125, 3281, 3378, 3784, 3864, 3972, 10192, 10426, 10521, 10525, 10575, 11163, 11665, 12100, 12104, 12321, 12493, 12505, 12595, 12769, 12843, 13125, 13496, 13764, 14035, 14129, 14350, 14641, 14647, 15379, 15568, 15613, 15617, 15621, 15622, 15625, 15626, 15627, 15628, 15633, 15634, 15641, 15656, 15688, 15689, 15975, 16245, 16375, 16765, 16783, 16815, 16875, 17325, 17689, 18723, 19629, 20485, 21870, 23490, 24385, 27783, 29160, 32855, 32865, 32895, 33495, 34425, 34991, 35726, 35782, 35933, 36457, 36549, 37179, 37875, 38427, 38640, 38856, 39288, 39363, 39366, 39369, 39372, 39784, 41323, 42898, 43689, 43692, 45363, 45365, 45366, 45369, 45684, 45873, 45957, 46613, 46630, 46633, 46641, 46642, 46644, 46648, 46661, 46662, 46665, 46668, 46673, 46851, 49277, 52168, 52493, 53865, 54378, 54476, 54872, 57645, 58921, 58978, 59032, 59044, 59048, 59056, 59265, 62564, 63904, 64036, 64522, 129285, 132527, 132567, 132587, 132797, 132854, 132860, 133563, 133815, 134375, 134455, 134459, 134462, 134799, 135025, 136079, 136892, 136945, 136948, 137459, 137623, 137759, 137789, 137790, 137815, 137837, 137838, 137879, 137899, 137942, 137948, 137955, 137962, 137964, 139455, 139965, 139966, 141288, 141659, 141750, 142296, 142875, 143857, 143982, 144298, 144527, 145314, 145345, 145652, 145656, 146202, 146337, 146461, 146517, 146758, 146979, 149364, 150246, 150435, 151250, 151272, 152406, 152685, 152946, 154275, 154950, 155464, 155838, 156247, 156251, 156253, 156254, 156267, 156275, 156277, 156278, 156282, 156315, 156439, 157393, 157394, 157447, 157453, 157459, 157465, 157468, 158922, 159045, 159049, 161045, 161050, 161052, 161055, 161059, 161728, 162408, 162450, 162720, 163216, 164583, 164775, 164842, 165834, 165842, 165848, 166347, 166374, 167274, 167281, 167377, 167384, 168750, 170343, 170472, 172743, 172773, 173259, 173426, 173784, 173849, 174195, 174298, 174652, 174682, 174762, 174793, 174802, 174820, 174825, 174982, 175203, 175273, 175274, 175329, 175373, 175560, 175623, 175644, 176215, 176249, 176435.

Some larger examples :