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economical n.  and  powerful n.

First numbers that belong to both sets:

1, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1000, 1024, 1125, 1152, 1225, 1296, 1323, 1331, 1369, 1372, 1568, 1600, 1681, 1728, 1849, 1944, 2000, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2592, 2744, 2809, 2916, 3087, 3125, 3136, 3200, 3375, 3456, 3481, 3721, 3888, 3969, 4000, 4096, 4489, 4608, 4913, 5000, 5041, 5184, 5329, 5488, 5625, 5832, 6075, 6125, 6241, 6272, 6400, 6561, 6859, 6889, 6912, 7776, 7921, 8000, 8192, 8575, 8748, 9261, 9409, 9604, 10000, 10125, 10201, 10368, 10609, 10648, 10816, 10952, 10976, 11236, 11449, 11552, 11664, 11881, 11907, 11979, 12167, 12321, 12500, 12544, 12769, 12800, 13225, 13448, 13456, 13689, 13824, 13924, 14161, 14283, 14641, 14792, 14884, 15125, 15129, 15376, 15488, 15552, 15625, 16000, 16129, 16384, 16641, 16807, 16875, 16928, 17161, 17496, 17576, 17672, 17689, 17956, 18225, 18432, 18496, 18769, 19208, 19321, 19652, 19683, 19773, 19881, 20000, 20164, 20736, 21025, 21125, 21296, 21316, 21609, 21632, 21904, 21952, 22201, 22472, 22707, 22801, 23104, 23328, 23409, 24025, 24389, 24649, 24964, 25000, 25088, 25281, 25600, 25921, 25947, 26244, 26569, 26896, 26912, 27436, 27556, 27648, 27783, 27848, 27889, 28125, 28561, 29241, 29403, 29584, 29768, 29791, 29929, 30375, 30625, 30752, 30976, 31104, 31329, 31684, 32000, 32041, 32761, 32768, 33275, 33489, 33856, 34225, 34992, 35152, 35344, 35721, 35912, 35937, 36125, 36481, 36864, 36963, 36992, 37249, 37636, 38416, 38809, 39304, 39601, 40000, 40328, 40401, 41067, 41209, 41472, 41503, 42025, 42592, 42632, 42849, 42875, 43264, 43808, 43904, 44217, 44521, 44944, 45125, 45369, 45387, 46208, 46225, 46656, 47089, 47961, 48668, 49729, 49923, 49928, 50000, 50176, 50625, 50653, 51200, 51529, 52441, 52488, 53792, 53824, 54289, 54675, 54872, 54925, 55112, 55225, 55296, 55696.

Some larger examples :