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centered triangular n.  and  Duffinian n.

First numbers that belong to both sets:

4, 64, 85, 235, 361, 901, 1135, 1219, 1585, 2461, 2839, 3385, 3979, 4135, 5311, 5491, 6049, 6241, 6835, 8551, 9481, 11485, 11749, 12559, 12835, 13969, 15151, 17335, 18649, 18985, 21421, 21781, 22879, 24385, 24769, 27949, 30889, 32635, 33079, 34885, 35344, 36271, 38161, 38641, 46201, 46729, 48331, 55585, 57919, 59701, 60301, 64585, 65209, 67099, 75601, 77635, 80389, 81085, 83191, 83899, 86041, 86761, 91885, 92629, 94879, 95635, 97921, 101011, 101791, 104941, 107335, 111385, 113851, 117181, 118021, 120559, 121411, 123985, 124849, 127459, 128335, 131869, 134551, 135451, 138169, 139081, 146485, 149311, 150259, 153121, 165835, 168841, 169849, 172891, 173911, 181135, 185329, 186385, 189571, 194941, 198199, 199291, 202585, 203689, 207019, 208135, 212629, 216031, 217171, 221761, 225235, 226399, 229909, 234631, 235819, 239401, 240601, 245431, 249085, 250309, 255235, 258961, 263971, 270301, 280585, 284491, 285799, 291061, 296371, 300385, 301729, 305779, 311221, 316711, 329239, 333469, 334885, 344881, 350659, 357949, 362359, 363835, 368281, 369769, 374251, 375751, 381781, 386335, 387859, 392449, 393985, 400159, 404821, 411079, 417385, 418969, 423739, 425335, 430141, 431749, 444721, 456229, 457885, 484789, 489919, 491635, 496801, 498529, 503731, 505471, 517735, 524809, 531931, 546319, 548131, 555409, 560899, 562735, 568261, 583129, 585001, 590635, 592519, 600085, 605791, 607699, 611524, 615361, 628885, 630829, 636679, 638635, 646489, 654391, 662341, 670339, 676369, 678385, 694621, 702811, 719335, 725581, 727669, 733951, 744481, 752959, 759349, 761485, 767911, 785179, 787351, 793885, 796069, 802639, 804835, 813649, 822511, 829189, 831421, 838135, 840379, 849385, 856171, 874399, 876691, 883585, 892819, 895135, 902101, 904429, 911431, 920809, 923161, 932599, 939709, 942085, 949231, 951619, 958801, 970831, 980509, 987799, 990235, 997561, 1000009, 1007371, 1009831, 1017229, 1027135, 1029619, 1047091, 1057141, 1059661, 1069771, 1077385, 1087579, 1097821, 1100389, 1108111, 1110691, 1118449, 1121041, 1128835, 1131439, 1149751, 1152379, 1162921, 1170859, 1181485, 1192159, 1194835, 1205569, 1224469, 1235335, 1238059, 1248985, 1257211, 1259959, 1279279, 1312741.

Some larger examples :