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ABA n.  and  balanced n.

First numbers that belong to both sets:

50, 384, 450, 578, 722, 968, 1922, 2738, 3528, 10082, 11858, 12168, 14450, 14792, 15138, 15625, 27783, 30258, 36501, 38880, 39762, 54450, 55112, 58482, 62658, 64800, 142578, 146882, 151250, 151959, 154568, 160178, 162450, 163592, 177608, 180000, 181202, 182408, 186050, 187272, 199712, 200978, 208658, 211250, 212552, 216482, 219122, 220448, 223112, 224450, 228488, 232562, 235298, 236672, 238050, 239432, 245000, 247808, 249218, 252050, 253472, 256328, 257762, 276768, 278258, 281250, 287282, 288800, 302642, 312050, 318402, 326432, 337842, 361250, 366368, 383688, 396050, 403202, 419528, 421824, 425042, 441800, 453152, 480200, 496008, 506018, 568178, 585362, 613832, 616050, 616137, 686792, 689138, 696200, 703298, 705894, 714984, 715208, 717602, 732050, 736898, 756450, 761378, 768800, 771282, 773768, 842402, 858050, 908552, 938450, 943938, 963272, 1016738, 1028178, 1042568, 2105352, 2171528, 2196608, 2209202, 2221832, 2226050, 2255688, 2298368, 2324168, 2345778, 2354450, 2358792, 2450898, 2486450, 2522258, 2562848, 2567378, 2571912, 2580992, 2608328, 2705138, 2714450, 2719112, 2723778, 2742482, 2832200, 2836962, 2856050, 2910897, 2913698, 2923362, 2942738, 2991458, 2996352, 3035648, 3055392, 3065288, 3070242, 3110018, 3130002, 3135008, 3155072, 3188646, 3276800, 3354050, 3495368, 3559112, 3564450, 3575138, 3601928, 3639602, 3672050, 3715538, 3731912, 3748322, 3770258, 3786752, 3797768, 3858642, 3864200, 3886472, 3897632, 3908808, 3931208, 3948050, 4004450, 4015778, 4072658, 4078368, 4135688, 4141442, 4452128, 4657352, 4755528, 5152050, 5307282, 5464818, 5497928, 5564448, 5705442, 5725728, 5827698, 5834528, 5951250, 6002500, 6013512, 6076098, 6223392, 6244578, 6544962, 6573938, 6741792, 6744273, 7212402, 7372800, 7884216, 7984008, 8056857, 8080200, 8225568, 8249922, 8569800, 8711138, 8719488, 8744562, 8853632, 8895762, 9014258, 9039752, 9107912, 9167762, 9253764, 9279432, 9313928, 9336408, 9357138, 9374450, 9391778, 9504800, 9653618, 9697608, 9777042, 9794738, 9883458, 9910152, 9919058, 9927968, 10116002, 10134002, 10170050, 10188098, 10296722, 10328853, 10333058, 10342152, 10378568, 10681442, 10718450, 10736978, 10746248, 10792658, 10867122, 11073218, 11110898, 11120328, 11129762, 11167538, 11205378, 11243282, 11300258, 11376450, 11385992, 11414642, 11452898, 11472050, 11587298, 11609679, 11625842, 11664450, 11936498, 12005000, 12073698, 12093362, 12122888, 12152450.

Some larger examples :