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frugal n.  and  powerful n.

First numbers that belong to both sets:

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192, 9409, 10000, 10125, 10201, 10368, 10609, 10976, 11449, 11664, 11881, 11907, 12167, 12500, 12544, 12769, 12800, 13824, 14641, 15552, 15625, 16000, 16129, 16384, 16807, 16875, 17161, 17496, 18225, 18769, 19208, 19321, 19683, 20000, 20736, 21609, 21952, 22201, 22801, 23328, 24389, 24649, 25000, 25088, 26244, 26569, 27783, 27889, 28125, 28561, 29791, 29929, 30375, 30625, 31104, 32000, 32041, 32761, 32768, 34992, 35721, 36481, 37249, 38416, 38809, 39601, 40000, 41472, 42875, 43904, 44521, 46656, 49729, 50000, 50625, 50653, 51529, 52441, 52488, 54289, 54675, 57121, 58081, 59049, 60025, 62208, 62500, 63001, 64000, 64827, 65536, 66049, 67228, 68921, 69169, 69984, 72361, 73441, 76729, 76832, 78125, 78732, 78961, 79507, 80000, 80089, 83349, 83521, 84375, 85849, 87808, 91125, 93312, 94249, 96721, 97969, 100000, 100352, 100467, 100489, 102400, 103823, 104976, 105125, 105625, 107163, 107584, 107648, 107653, 107811, 108241, 109503, 109561, 109744, 110224, 110592, 110889, 111392, 112225, 113569, 114244, 117128, 117649, 118336, 119072, 119164, 120125, 120409, 121203, 121801, 122825, 123008, 123201, 123823, 124416, 124609, 125000, 126025, 126736, 128000, 128547, 128881, 130321, 131072, 131769, 132651, 133225, 134456, 134689, 135424, 136107, 136161, 137641, 139129, 139968, 140608, 140625, 141376, 143641, 143648, 143883, 146689, 147456, 147968, 148877, 149769, 150544, 151263, 151321, 151875, 153125, 153664, 156025, 157216, 157464, 157609, 160000, 160801, 161051, 161312, 164025, 165888, 166375, 167281, 168507, 170368, 170528, 170569, 171125, 171475, 172225, 175232, 175561, 175616, 177147, 177241, 177957, 178929, 179776, 180625, 181447, 182329, 184832, 185193, 185761, 186003, 186624, 187489, 192721, 194481, 194672, 195112, 196249, 198025, 199712, 200000, 200704, 201601, 202612, 204363, 204800, 205379, 208849, 209952, 210125, 210681, 212521, 213867.

Some larger examples :