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frugal n.  and  interprime n.

First numbers that belong to both sets:

625, 1215, 1681, 1715, 1875, 4096, 10443, 10976, 11125, 12500, 12544, 14739, 15104, 16875, 18605, 20992, 21141, 21952, 26047, 27097, 28431, 31347, 34375, 36015, 43923, 46656, 50301, 70713, 72171, 83349, 90625, 91854, 93750, 94041, 96875, 98415, 100359, 102016, 102912, 103058, 104375, 104704, 105219, 108135, 109056, 109107, 109952, 110592, 111051, 114244, 121875, 122451, 123875, 126117, 126736, 128625, 131061, 131250, 131463, 131712, 133837, 135351, 138267, 139968, 141426, 142805, 143125, 143613, 144669, 145408, 147987, 148959, 149769, 150577, 151263, 153856, 159375, 163607, 164608, 166698, 172773, 174231, 175689, 179375, 180075, 183708, 185895, 186624, 186880, 193671, 194481, 194816, 195112, 195533, 197559, 203264, 204800, 206486, 210625, 211259, 219520, 220401, 226719, 231093, 231822, 235298, 240267, 240896, 248320, 250563, 254893, 255507, 258795, 262205, 263169, 270625, 274432, 276125, 277749, 279207, 281250, 283136, 283581, 285605, 289375, 290625, 293378, 300000, 301467, 306250, 307461, 311296, 311875, 314432, 315657, 316928, 318647, 319333, 321489, 321602, 322763, 330237, 334611, 336069, 344605, 346875, 351875, 354375, 354631, 360448, 366875, 370625, 381759, 384375, 387099, 388962, 389375, 403137, 405224, 406250, 406567, 425845, 428750, 431875, 440067, 442223, 449307, 459375, 463393, 466432, 468750, 472392, 478953, 485514, 501126, 507392, 507904, 509571, 511413, 514087, 518319, 538731, 542882, 546875, 555131, 565625, 571875, 581013, 589761, 597051, 599781, 604341, 605799, 611631, 618921, 623295, 638666, 639333, 641067, 643125, 656250, 663552, 664119, 671409, 685989, 689210, 703125, 708588, 723897, 732305, 772011, 786432, 787320, 796322, 820587, 822649, 835210, 840350, 842863, 852818, 868239, 872263, 876987, 878080, 885735, 888191, 909375, 933849, 957999, 966656, 972405, 984375, 991613, 995328, 1001375, 1001875, 1002252, 1003686, 1005487, 1005824, 1008420, 1009408, 1010394, 1010821, 1012625, 1014049, 1019264, 1020125, 1020160, 1020416, 1020425, 1020625, 1021329, 1022767, 1023104, 1023323, 1024338, 1032192, 1033479, 1036625, 1036875, 1037056, 1037500, 1038825, 1040384, 1041157, 1043750, 1047573, 1047757, 1049031, 1049760, 1052919, 1053824, 1054377, 1057536, 1058751, 1059723, 1060864, 1061875, 1063168.

Some larger examples :