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economical n.  and  perfect power

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, 1225, 1296, 1331, 1369, 1600, 1681, 1728, 1849, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2744, 2809, 2916, 3125, 3136, 3375, 3481, 3721, 3969, 4096, 4489, 4913, 5041, 5184, 5329, 5625, 5832, 6241, 6400, 6561, 6859, 6889, 7776, 7921, 8000, 8192, 9261, 9409, 9604, 10000, 10201, 10609, 10648, 10816, 11236, 11449, 11664, 11881, 12167, 12321, 12544, 12769, 13225, 13456, 13689, 13824, 13924, 14161, 14641, 14884, 15129, 15376, 15625, 16129, 16384, 16641, 16807, 17161, 17576, 17689, 17956, 18225, 18496, 18769, 19321, 19683, 19881, 20164, 20736, 21025, 21316, 21609, 21904, 21952, 22201, 22801, 23104, 23409, 24025, 24389, 24649, 24964, 25281, 25600, 25921, 26244, 26569, 26896, 27556, 27889, 28561, 29241, 29584, 29791, 29929, 30625, 30976, 31329, 31684, 32041, 32761, 32768, 33489, 33856, 34225, 35344, 35721, 35937, 36481, 36864, 37249, 37636, 38416, 38809, 39304, 39601, 40000, 40401, 41209, 42025, 42849, 42875, 43264, 44521, 44944, 45369, 46225, 46656, 47089, 47961, 49729, 50176, 50625, 50653, 51529, 52441, 53824, 54289, 54872, 55225, 55696, 56169, 57121, 58081, 58564, 59049, 59319, 59536, 60025, 61504, 62001, 62500, 63001, 64000, 65536, 66049, 67081, 68121, 68921, 69169, 70225, 71289, 71824, 72361, 73441, 73984, 75625, 76729, 77841, 78125, 78961, 79507, 80089, 80656, 82369, 82944, 83521, 84681, 85184, 85264, 85849, 87025, 87616, 88209, 90601, 91125, 92416, 93025, 94249, 96721, 97336, 97969, 99856, 100000, 100489, 101761, 102400, 103041, 103823, 104329, 104976, 105625, 106276, 106929, 107584, 108241, 109561, 110224, 110592, 110889, 111556, 112225, 112896, 113569, 114244, 114921, 116281, 117649, 118336, 119716, 120409, 121801, 122500, 123201, 123904, 124609, 125000, 126025, 126736, 128164, 128881, 129600, 130321, 131044, 131072, 131769, 132651, 133225, 134689.

Some larger examples :