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Duffinian n.  and  frugal n.

First numbers that belong to both sets:

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1250, 1331, 1369, 1681, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4375, 4489, 4802, 4913, 5041, 5103, 5329, 6241, 6561, 6859, 6889, 7203, 7921, 8192, 9409, 10000, 10082, 10201, 10375, 10443, 10449, 10609, 10625, 10633, 10658, 11045, 11449, 11664, 11767, 11881, 12005, 12125, 12167, 12482, 12691, 12769, 12800, 12943, 13125, 13778, 13851, 14045, 14375, 14641, 14749, 14823, 15123, 15309, 15463, 15625, 15842, 16121, 16129, 16281, 16384, 16807, 17161, 17405, 17739, 18179, 18225, 18605, 18769, 18818, 19197, 19208, 19321, 19375, 19683, 20000, 20237, 20577, 20667, 20736, 20923, 21609, 22201, 22445, 22599, 22801, 22981, 23125, 23571, 23763, 24353, 24367, 24389, 24649, 25039, 25088, 25205, 25625, 26047, 26244, 26411, 26569, 26645, 26875, 26973, 27097, 27783, 27889, 28431, 28561, 29282, 29375, 29791, 29929, 30527, 30625, 31205, 31250, 31347, 32041, 32761, 32768, 33125, 34375, 34391, 34445, 35287, 36481, 37249, 37303, 38125, 38416, 38809, 39601, 39605, 40000, 40625, 40817, 41553, 41875, 43923, 44469, 44521, 45619, 45625, 45927, 46656, 46875, 47045, 48013, 48223, 48843, 49729, 50625, 50653, 51529, 51875, 52441, 53125, 53217, 54289, 55223, 55447, 57121, 57122, 57591, 58081, 59049, 60025, 60625, 62500, 63001, 64827, 65536, 65863, 66049, 67797, 68921, 69169, 69629, 70713, 71875, 72361, 73441, 74431, 76729, 78125, 78961, 79507, 80089, 80919, 83521, 84035, 85169, 85683, 85849, 88837, 89373, 94041, 94249, 96721, 97969, 100489, 100625, 100949, 101251, 101331, 101375, 101875, 102303, 102400, 102487, 102541, 102625, 102875, 102973, 103058, 103243, 103247, 103259, 103375, 103499, 103823, 103933, 104375, 104882, 104976, 105125, 105149, 105219, 105625, 105861, 106097, 106625, 106673, 106677, 107125, 107359, 107584, 107648, 107875, 107909, 107911, 108125, 108151, 108241, 108578, 108731, 109091, 109375, 109443, 109561, 109625, 110079, 110125, 110224, 110375, 110473, 110875, 111005.

Some larger examples :