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arithmetic n.  and  frugal n.

First numbers that belong to both sets:

125, 343, 1029, 1215, 1331, 1369, 1715, 1849, 2187, 2197, 3125, 3721, 4374, 4489, 4913, 5329, 6241, 6859, 9375, 9409, 10112, 10375, 10609, 10633, 10935, 10976, 10985, 11045, 11125, 11163, 11421, 11875, 11881, 12125, 12167, 12500, 12691, 12879, 12943, 13467, 13568, 13718, 13824, 14045, 14063, 14336, 14337, 14641, 14739, 14749, 15309, 15379, 15987, 16121, 16129, 16807, 16875, 17253, 17405, 18125, 18176, 18179, 18605, 18723, 18750, 19321, 20169, 20237, 20577, 20923, 21627, 21870, 21875, 22445, 22784, 22801, 22981, 24057, 24334, 24353, 24389, 24565, 24649, 25039, 25205, 26047, 26569, 26645, 27097, 27783, 28125, 28227, 28431, 28469, 29791, 29889, 30208, 30375, 30527, 30618, 31205, 31423, 32761, 33271, 33614, 34295, 34375, 34391, 34445, 34992, 36501, 36875, 37179, 37249, 37303, 39601, 39605, 40448, 40625, 41553, 42875, 43687, 43750, 43923, 44521, 45619, 47045, 48013, 49375, 49729, 50301, 50421, 50653, 52441, 53125, 55625, 58081, 59375, 59582, 60507, 60835, 63423, 64827, 65625, 65863, 67797, 68921, 69629, 69984, 70713, 71875, 73167, 73205, 73441, 76545, 76729, 78125, 79507, 80089, 80919, 84035, 84375, 85169, 89373, 89667, 90625, 94041, 94249, 96875, 97969, 100359, 100467, 100499, 100602, 100842, 100845, 100949, 101125, 101185, 101251, 101331, 101375, 101817, 101888, 102487, 102625, 102789, 102875, 102885, 103125, 103173, 103247, 103259, 103335, 103375, 103499, 103625, 103823, 103936, 104615, 104733, 104875, 105125, 105149, 105301, 106097, 106240, 106625, 106673, 107125, 107163, 107359, 107375, 107649, 107653, 107875, 107909, 108135, 108151, 108731, 109091, 109107, 109350, 109503, 109561, 109625, 109744, 110125, 110336, 110375, 110464, 110473, 110625, 110875, 110894, 111005, 111537, 111747, 111875, 111909, 112023, 112789, 112903, 112999, 113375, 113481, 113533, 113569, 113627, 113875, 113920, 114005, 114875, 114905, 114944, 115169, 115351, 115591, 115625, 115911, 115943, 116125, 116397, 116441, 116603, 116608, 117077, 117113, 117125, 117625, 117855, 118125, 118375, 118459, 118579, 118750, 118803, 118815, 119021, 119125.

Some larger examples :