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Duffinian n.  and  triangular n.

First numbers that belong to both sets:

21, 36, 55, 171, 253, 325, 351, 595, 741, 903, 1081, 1225, 1711, 1953, 2485, 2775, 3081, 3403, 3655, 5671, 7021, 7381, 8001, 8515, 8911, 10153, 11325, 11935, 13203, 13861, 14365, 15931, 20503, 21115, 23005, 23653, 23871, 24753, 25651, 28441, 29403, 31375, 32131, 33411, 34453, 37675, 38781, 41616, 44551, 45451, 48205, 52003, 52975, 53301, 55945, 58653, 60031, 64261, 65341, 65703, 67161, 68635, 71631, 73153, 74305, 74691, 79401, 81003, 82621, 87571, 92665, 96141, 97903, 108811, 110215, 114481, 115921, 118341, 120295, 121771, 126253, 127765, 132355, 133903, 138601, 140185, 144991, 147153, 153181, 158203, 159895, 160461, 167331, 171991, 179101, 181503, 186355, 191271, 193753, 195625, 201295, 208981, 216811, 224785, 232903, 235641, 243253, 249571, 258121, 260281, 263901, 266815, 275653, 290703, 293761, 303031, 305371, 314821, 322003, 331705, 334153, 341551, 351541, 354061, 361675, 371953, 374545, 375411, 392941, 403651, 406351, 410871, 417241, 425503, 428275, 429201, 436645, 440391, 459361, 462241, 470935, 474825, 482653, 498501, 506521, 518671, 526851, 530965, 543403, 547581, 551775, 555985, 568711, 584821, 590241, 603351, 607753, 621055, 625521, 634501, 651511, 652653, 661825, 671061, 679195, 689725, 693253, 703891, 747253, 750925, 752151, 776881, 791911, 795691, 796953, 807085, 822403, 826255, 837865, 848253, 853471, 869221, 889111, 901153, 917335, 933661, 937765, 983503, 1000405, 1004653, 1011753, 1017451, 1034641, 1046181, 1063611, 1073845, 1075311, 1091503, 1098903, 1104841, 1122751, 1127251, 1140805, 1145341, 1159003, 1181953, 1183491, 1214461, 1233235, 1237951, 1239525, 1252153, 1256905, 1258491, 1290421, 1309771, 1316253, 1322751, 1329265, 1334161, 1348903, 1353835, 1355481, 1368685, 1375311, 1381953, 1388611, 1408681, 1413721, 1442451, 1449253, 1454365, 1490401, 1511191, 1525131, 1532125, 1553203, 1558495, 1560261, 1574425, 1595791, 1617301, 1638955, 1644391, 1660753, 1666225, 1682695, 1688203, 1690041, 1697403, 1704781, 1710325, 1727011, 1749385, 1756875, 1764381, 1771903, 1817371, 1823095, 1840321, 1846081, 1863415, 1886653, 1902225, 1910035, 1915903, 1933561, 1957231, 1963171, 1965153, 1973091, 1997001, 2037171, 2045253, 2053351, 2077741, 2083861, 2126953, 2151775, 2158003.

Some larger examples :