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Duffinian n.  and  powerful n.

First numbers that belong to both sets:

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 225, 243, 256, 289, 324, 343, 361, 392, 400, 484, 512, 529, 576, 625, 676, 729, 784, 800, 841, 900, 961, 968, 1024, 1089, 1156, 1225, 1296, 1331, 1352, 1369, 1444, 1600, 1681, 1849, 1936, 2025, 2048, 2116, 2187, 2197, 2209, 2304, 2312, 2401, 2500, 2601, 2704, 2809, 2888, 2916, 3025, 3087, 3125, 3136, 3267, 3364, 3481, 3600, 3721, 3844, 3872, 4096, 4225, 4232, 4356, 4489, 4624, 4761, 4913, 5041, 5184, 5329, 5408, 5476, 5625, 5776, 6075, 6125, 6241, 6272, 6400, 6561, 6724, 6728, 6859, 6889, 7225, 7396, 7569, 7688, 7744, 7803, 7921, 8100, 8192, 8281, 8464, 8836, 9025, 9216, 9248, 9261, 9409, 10000, 10201, 10404, 10609, 10816, 10952, 11236, 11449, 11552, 11664, 11881, 12100, 12167, 12769, 12800, 13225, 13448, 13456, 13924, 14161, 14283, 14400, 14641, 14792, 14884, 15125, 15129, 15488, 15625, 16129, 16384, 16807, 16900, 16928, 17161, 17424, 17672, 17956, 18225, 18496, 18769, 19044, 19208, 19321, 19600, 19683, 19881, 20000, 20164, 20449, 20736, 21025, 21316, 21609, 21632, 21904, 22201, 22472, 22500, 22707, 22801, 23104, 23409, 24389, 24649, 24964, 25088, 25281, 25600, 26244, 26569, 26896, 26912, 27225, 27556, 27783, 27848, 27889, 28561, 28900, 29403, 29584, 29768, 29791, 29929, 30276, 30625, 30752, 30976, 31329, 31684, 32041, 32400, 32761, 32768, 33275, 33856, 34225, 34969, 35344, 35912, 36100, 36125, 36481, 36864, 37249, 37636, 38416, 38809, 39601, 40000, 40328, 40804, 41209, 41616, 42025, 42436, 42632, 42849, 43264, 43808, 44521, 44944, 45125, 45369, 45387, 45796, 46208, 46225, 46656, 47089, 47524, 48400, 48841, 49729, 49928, 50176, 50625, 50653, 51076, 51529, 52441, 52900, 53792, 53824, 54289, 55112, 55225, 55696, 57121, 57600, 57967, 58081, 58564, 59049, 59168, 59319, 59536, 59643, 60025.

Some larger examples :