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ABA n.  and  Duffinian n.

First numbers that belong to both sets:

8, 32, 50, 64, 81, 98, 128, 242, 324, 338, 392, 512, 578, 722, 800, 968, 1024, 1029, 1058, 1250, 1352, 1682, 1922, 2048, 2187, 2312, 2450, 2500, 2738, 2888, 3362, 3698, 3872, 4232, 4418, 4802, 5184, 5408, 5618, 6050, 6272, 6591, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 10082, 10658, 10952, 11552, 12482, 12800, 13448, 13778, 14450, 14792, 15309, 15488, 15625, 15842, 16384, 16562, 16928, 17672, 18050, 18818, 19208, 20000, 20402, 20577, 21218, 21632, 22472, 22898, 23762, 25088, 25538, 26244, 26450, 26912, 27783, 27848, 28322, 29282, 29768, 30752, 31250, 32258, 32768, 34322, 35912, 37538, 38642, 40000, 40328, 40898, 42050, 42632, 43808, 44402, 45602, 46208, 46875, 49298, 49928, 53138, 53792, 55112, 55778, 57122, 58564, 59049, 59168, 59858, 61250, 63368, 64082, 65522, 66248, 67712, 68450, 69938, 70688, 72962, 74498, 75272, 77618, 79202, 81608, 82418, 82944, 84035, 84050, 84872, 86528, 89042, 89373, 89888, 91592, 92450, 94178, 95048, 96800, 97682, 99458, 102152, 103058, 104882, 107648, 108578, 110450, 111392, 113288, 114242, 114244, 116162, 117128, 119072, 120050, 122018, 123008, 126002, 128018, 129032, 131072, 132098, 135200, 137288, 138338, 140450, 143648, 144722, 146882, 147968, 150152, 151959, 153458, 153664, 154568, 157922, 160178, 161312, 163592, 164738, 167042, 170528, 171698, 174050, 175232, 177147, 177608, 177957, 178802, 181202, 182408, 184832, 186050, 188498, 193442, 195938, 197192, 199712, 200978, 202500, 203522, 204800, 208658, 211250, 212552, 215168, 216482, 219122, 220448, 223112, 224450, 227138, 228488, 231200, 232562, 235298, 236672, 238521, 239432, 240818, 243602, 247808, 249218, 252050, 253472, 256328, 257762, 260642, 262088, 262144, 266450, 269378, 270848, 278258, 279752, 282752, 287282, 288800, 291848, 293378, 297992, 301088, 302642, 305762, 307328, 310472, 312050, 315218, 316808, 320000, 321602, 324818, 326432, 329672, 331298, 331776, 334084, 334562, 339488, 341138, 344450, 351122, 352947, 354482, 356168, 359552, 361250, 364658, 366368, 371522, 374978, 376712.

Some larger examples :