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Achilles n.  and  Duffinian n.

First numbers that belong to both sets:

392, 800, 968, 1352, 2312, 2888, 3087, 3267, 3872, 4232, 5408, 6075, 6125, 6272, 6728, 7688, 7803, 9248, 10952, 11552, 12800, 13448, 14283, 14792, 15125, 15488, 16928, 17672, 19208, 20000, 21632, 22472, 22707, 25088, 26912, 27783, 27848, 29403, 29768, 30752, 33275, 35912, 36125, 40328, 42632, 43808, 45125, 45387, 46208, 49928, 53792, 55112, 57967, 59168, 59643, 61731, 63368, 64827, 65219, 66125, 66248, 67712, 70227, 70688, 75272, 75843, 81608, 84872, 86528, 89888, 91592, 93987, 95048, 96800, 99127, 102152, 105125, 107648, 111392, 113288, 117128, 119072, 120125, 123008, 123823, 128547, 129032, 135200, 136107, 137288, 143648, 147968, 150152, 151875, 154568, 161312, 163592, 170528, 171125, 175232, 177608, 177957, 182408, 184832, 186003, 197192, 199712, 204363, 204800, 210125, 212552, 213867, 215168, 220448, 223112, 224939, 228488, 231125, 231200, 236672, 239432, 240737, 247808, 253472, 256328, 262088, 264627, 265837, 268119, 270848, 275427, 276125, 279752, 282752, 288463, 288800, 291848, 297992, 300125, 301088, 307328, 309123, 310472, 316808, 320000, 326432, 329623, 329672, 339488, 344763, 351125, 356168, 359552, 366368, 376712, 378125, 380192, 384659, 390728, 395307, 397832, 401408, 408483, 408608, 412232, 419528, 420175, 423200, 430592, 434312, 435125, 445568, 455877, 456968, 463347, 464648, 465125, 468512, 476288, 480491, 488072, 492075, 500000, 504008, 506763, 512072, 516128, 528125, 528392, 536787, 549152, 553352, 555579, 561125, 574592, 576583, 578888, 587528, 591872, 594473, 596183, 599427, 600608, 613832, 618272, 627200, 630125, 631688, 632043, 634207, 640712, 645248, 654368, 658952, 666125, 668168, 672800, 682112, 682587, 686792, 700928, 704099, 710432, 715208, 715563, 724808, 729632, 735075, 739328, 750141, 753003, 753992, 757687, 771147, 773768, 780125, 783752, 788768, 793117, 798848, 803912, 808083, 814088, 830297, 834632, 845883, 850208, 860672, 861125, 865107, 865928, 876488, 881792, 892143, 892448, 903125, 908552, 913952, 930248, 944163, 946688, 957728, 963272, 974408, 984987, 990125, 991232, 996872, 1013888, 1025312, 1031048, 1042568, 1047843, 1048352, 1059968, 1077512, 1083392, 1095200, 1113032, 1119008.

Some larger examples :