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Achilles n.  and  economical n.

First numbers that belong to both sets:

1125, 1152, 1323, 1372, 1568, 1944, 2000, 2592, 3087, 3200, 3456, 3888, 4000, 4608, 5000, 5488, 6075, 6125, 6272, 6912, 8575, 8748, 10125, 10368, 10952, 10976, 11552, 11907, 11979, 12500, 12800, 13448, 14283, 14792, 15125, 15488, 15552, 16000, 16875, 16928, 17496, 17672, 18432, 19208, 19652, 19773, 20000, 21125, 21296, 21632, 22472, 22707, 23328, 25000, 25088, 25947, 26912, 27436, 27648, 27783, 27848, 28125, 29403, 29768, 30375, 30752, 31104, 32000, 33275, 34992, 35152, 35912, 36125, 36963, 36992, 40328, 41067, 41472, 41503, 42592, 42632, 43808, 43904, 44217, 45125, 45387, 46208, 48668, 49923, 49928, 50000, 51200, 52488, 53792, 54675, 54925, 55112, 55296, 57967, 59168, 59643, 61731, 61952, 62208, 63368, 64827, 65219, 66125, 67228, 67712, 69984, 70227, 70304, 70688, 73728, 75272, 75843, 76832, 78608, 78732, 80000, 83349, 84375, 86528, 87723, 87808, 89888, 93312, 93987, 97556, 99127, 100352, 100467, 102152, 105125, 107163, 107648, 107653, 107811, 108000, 109503, 109744, 111132, 111392, 112500, 115200, 117128, 119072, 119164, 120125, 121203, 121500, 122825, 123008, 123823, 124416, 127008, 128000, 128547, 129032, 134456, 135000, 136107, 137200, 137288, 139968, 143648, 143883, 144000, 145800, 147968, 148176, 150152, 151263, 151875, 153125, 154568, 156800, 157216, 161312, 162000, 165375, 165888, 168507, 169344, 170368, 170528, 171125, 171475, 171500, 172800, 172872, 175232, 177608, 177957, 180000, 181447, 182408, 184832, 186003, 190512, 194400, 194672, 196000, 197192, 197568, 199712, 200000, 202612, 204363, 204800, 209952, 210125, 212552, 213867, 214375, 215168, 218700, 219488, 219501, 220448, 221184, 222264, 223112, 224939, 225000, 225792, 228488, 231125, 231525, 233523, 236672, 239432, 240737, 243000, 245000, 247808, 253125, 253472, 254043, 256000, 256328, 259200, 259308, 262088, 264627, 265837, 268119, 268912, 270000, 270848, 273375, 274400, 275427, 275684, 276125, 281216, 282752, 285768, 286443, 288000, 288463, 291848, 294912, 296352, 297675, 297992, 300125, 301088, 304175, 307328, 309123, 310472, 312500, 314928, 316808, 318028, 320000, 320787, 323433.

Some larger examples :