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congruent n.  and  frugal n.

First numbers that belong to both sets:

125, 343, 1029, 1215, 1280, 1536, 1701, 1792, 2197, 3125, 3584, 3645, 4374, 4375, 5103, 9375, 10112, 10125, 10206, 10375, 10496, 10624, 10935, 10976, 10985, 11045, 11125, 11421, 11767, 11776, 12005, 12032, 12125, 12167, 12416, 12500, 12879, 12943, 13125, 13568, 13718, 13824, 14045, 14063, 14336, 14375, 14406, 14749, 14823, 15309, 15463, 15616, 15872, 16767, 16807, 17253, 17405, 17496, 17739, 17920, 18125, 18176, 18179, 18605, 18750, 19197, 19375, 19663, 20224, 20237, 20480, 21141, 21870, 21952, 22016, 22445, 22599, 22981, 23125, 23571, 24064, 24334, 24367, 24389, 24565, 24576, 25039, 25205, 25625, 26047, 26645, 26973, 27783, 28125, 28431, 28469, 28672, 29375, 29791, 29889, 30208, 30375, 30527, 31104, 31205, 31213, 31423, 32000, 32805, 33125, 33271, 33614, 34263, 34295, 34304, 34375, 34391, 34445, 35287, 36015, 36352, 36501, 37303, 38125, 38637, 39366, 39605, 40448, 40625, 42496, 43687, 43750, 43904, 44375, 44469, 45927, 47045, 48013, 48223, 49375, 49664, 50000, 50301, 50421, 50653, 51759, 53125, 55223, 55447, 57344, 57591, 59375, 59582, 63423, 65863, 67228, 67797, 68921, 69629, 69984, 73167, 73205, 74431, 78125, 80919, 81920, 83349, 84375, 85169, 85293, 87808, 88837, 89373, 90625, 91125, 91854, 93750, 98304, 98415, 98441, 100359, 100602, 100625, 100736, 100845, 100949, 101120, 101125, 101375, 101632, 101888, 102303, 102487, 102541, 102789, 102885, 102973, 103125, 103173, 103247, 103335, 103375, 103518, 103680, 103750, 103808, 103823, 103933, 103936, 104375, 104615, 104733, 104960, 105125, 105149, 105301, 105344, 105705, 105728, 105856, 105861, 106240, 106677, 107125, 107359, 107375, 107392, 107653, 107776, 107909, 107911, 108032, 108045, 108125, 108135, 108151, 108621, 108864, 109056, 109350, 109375, 109503, 109952, 110079, 110080, 110125, 110336, 110375, 110446, 110464, 110894, 111005, 111104, 111909, 112000, 112023, 112128, 112384, 112509, 112768, 112789, 112847, 112903, 112999, 113024, 113125, 113375, 113533, 113536, 113920, 113967, 114005, 114176, 114375, 114453, 114688, 114875, 115169, 115182, 115351.

Some larger examples :