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ABA n.  and  frugal n.

First numbers that belong to both sets:

128, 512, 1024, 1029, 1215, 1250, 1458, 1536, 2048, 2187, 4374, 4802, 8192, 10082, 10125, 10240, 10368, 10658, 12288, 12482, 12800, 13122, 13778, 14739, 15309, 15625, 15842, 16384, 17496, 18818, 19208, 20000, 20577, 23328, 24576, 25088, 26244, 27783, 29282, 31250, 32768, 36501, 40000, 41472, 46875, 49152, 52488, 57122, 59049, 73167, 76832, 80000, 84035, 89373, 93312, 93750, 98304, 100352, 101250, 103058, 104882, 106496, 107648, 107811, 108578, 111392, 114242, 114244, 114688, 116162, 117128, 118098, 119072, 120050, 123008, 125000, 126002, 128625, 131072, 132098, 138338, 139968, 143648, 144722, 146882, 147968, 151959, 153458, 153664, 157922, 160178, 161312, 163840, 165888, 167042, 170528, 171698, 175232, 177147, 177957, 184832, 188498, 192000, 193442, 195938, 199712, 200978, 204800, 206763, 209952, 215168, 219122, 220448, 227138, 228488, 229376, 235298, 236672, 238521, 240818, 243602, 249218, 253472, 257762, 260642, 262144, 269378, 270848, 273375, 278258, 279936, 281250, 282752, 287282, 293378, 294912, 295245, 301088, 302642, 307328, 311469, 315218, 320000, 321602, 328050, 331776, 334084, 334562, 351122, 352947, 354482, 359552, 371522, 373248, 374978, 375000, 385442, 388962, 392498, 397953, 401408, 403202, 417698, 419904, 425042, 428738, 430592, 436178, 445568, 446631, 458882, 468512, 472392, 474338, 476288, 482162, 491520, 492032, 498002, 500000, 506018, 518162, 521284, 524288, 526848, 542882, 546875, 547058, 555579, 559682, 574592, 585362, 590490, 598418, 616137, 620498, 633938, 640000, 642978, 645248, 647522, 652082, 663552, 665858, 668168, 680943, 682112, 689138, 700928, 703298, 705894, 717602, 722402, 736898, 750141, 751538, 761378, 766322, 781250, 786432, 796322, 798848, 805255, 819200, 821762, 826898, 837218, 839808, 852818, 860672, 868562, 873842, 881792, 902289, 905858, 911250, 913952, 916658, 932978, 937024, 941192, 946688, 954962, 982802, 985527, 1005362, 1011042, 1013888, 1025312, 1031048, 1033922, 1036800, 1042568, 1048352, 1048576, 1051250, 1057058, 1062882, 1073733, 1074578, 1077512, 1083392, 1092242, 1104098, 1113032, 1116018, 1119364, 1119744, 1125000, 1128002, 1131008, 1143072, 1146098, 1149128, 1158242, 1167051, 1167392, 1173512, 1179648, 1182722.

Some larger examples :