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alternating n.  and  frugal n.

First numbers that belong to both sets:

125, 256, 343, 729, 1250, 1458, 2187, 6561, 10125, 10985, 12125, 12167, 12769, 12943, 14749, 16121, 16129, 16767, 18125, 18723, 18769, 21870, 30527, 32761, 36501, 38125, 41472, 50301, 65610, 72361, 76545, 76729, 78125, 78961, 96721, 101250, 101632, 103058, 105856, 109056, 109696, 121014, 121216, 121250, 121670, 125056, 125696, 127232, 127616, 129654, 143872, 147456, 161838, 163296, 165632, 169216, 169472, 181250, 210125, 210983, 212341, 212521, 214369, 214569, 216763, 218125, 230121, 230187, 234981, 236927, 238383, 238521, 250563, 252105, 252963, 256907, 258125, 276125, 278125, 290163, 290521, 292329, 298161, 303616, 323072, 329216, 341056, 349696, 367416, 381250, 389632, 416745, 418147, 430923, 434581, 436921, 438125, 438129, 452709, 452929, 454167, 458125, 458329, 476561, 498125, 503296, 547058, 581632, 614125, 614547, 618921, 630125, 656903, 658503, 678125, 690363, 703298, 705894, 781250, 814323, 850743, 876365, 876987, 878125, 905418, 905858, 1010125, 1012581, 1016309, 1018349, 1018367, 1018961, 1030301, 1030563, 1030903, 1032507, 1034125, 1038361, 1052363, 1052761, 1054729, 1054947, 1056321, 1058125, 1058529, 1070503, 1072561, 1074303, 1074789, 1076125, 1078125, 1078727, 1090341, 1090581, 1092125, 1092701, 1092727, 1094125, 1096303, 1098125, 1212125, 1212327, 1212505, 1212541, 1214303, 1214563, 1214941, 1216701, 1218125, 1230309, 1232125, 1236141, 1238125, 1250721, 1250921, 1252161, 1252503, 1256921, 1270129, 1270161, 1270387, 1270565, 1272105, 1272187, 1276303, 1278125, 1278361, 1290709, 1292767, 1292769, 1294507, 1294947, 1294949, 1296149, 1298125, 1298941, 1410129, 1412185, 1412927, 1414189, 1414503, 1414989, 1416521, 1416707, 1418125, 1418305, 1430541, 1434769, 1436141, 1450547, 1450703, 1452549, 1458125, 1458729, 1474767, 1478125, 1478581, 1478741, 1492121, 1492749, 1494341, 1496509, 1496541, 1498581, 1610361, 1618125, 1630729, 1632589, 1638125, 1638363, 1638505, 1654101, 1656147, 1656509, 1658125, 1658181, 1658961, 1670729, 1672569, 1676927, 1678125, 1692705, 1692981, 1694763, 1694925, 1698721, 1812127, 1816103, 1818125, 1818929, 1830125, 1832549, 1852321, 1852389, 1856749, 1856763, 1858107, 1858125, 1858707, 1858789, 1874161, 1878125, 1878541, 1892727, 1894389, 1896129, 1898125, 2101250, 2107616, 2109894, 2121472, 2125210, 2127616, 2141696, 2143690, 2147072, 2147634, 2181250, 2181632, 2181654.

Some larger examples :