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hexagonal n.  and  pernicious n.

First numbers that belong to both sets:

6, 28, 66, 91, 276, 496, 861, 1225, 1431, 1540, 1653, 1770, 1891, 2850, 3160, 3655, 4186, 4371, 4560, 4753, 4950, 5151, 5356, 8128, 10296, 10585, 11476, 12403, 13366, 13695, 14365, 16110, 16836, 17205, 18721, 19110, 21528, 22791, 24531, 24976, 25425, 25878, 26335, 26796, 27730, 30135, 35778, 43660, 49141, 49770, 51040, 54285, 56953, 60378, 61776, 62481, 63190, 65341, 67528, 68265, 70500, 72010, 72771, 73536, 74305, 75078, 81406, 82215, 85491, 87990, 88831, 97461, 98346, 101025, 109278, 112101, 114003, 114960, 117855, 118828, 119805, 120786, 124750, 137026, 141246, 142311, 143380, 144453, 148785, 161028, 162165, 164451, 165600, 167910, 173755, 176121, 184528, 188191, 189420, 190653, 193131, 198135, 200661, 210925, 217470, 218791, 220116, 221445, 234955, 237705, 243253, 251695, 263175, 264628, 267546, 271953, 273430, 274911, 279378, 282376, 296065, 297606, 303810, 306936, 319600, 324415, 329266, 339076, 340725, 344035, 345696, 350703, 357435, 359128, 364231, 372816, 376278, 378015, 379756, 381501, 390286, 400960, 402753, 406351, 431985, 439453, 441330, 450775, 458403, 481671, 485605, 489555, 491536, 493521, 505515, 507528, 509545, 513591, 515620, 517653, 519690, 521731, 536130, 544446, 546535, 563391, 574056, 584821, 586986, 591328, 597871, 611065, 617716, 619941, 633375, 635628, 651511, 667590, 669903, 674541, 679195, 693253, 695610, 697971, 700336, 702705, 717003, 724206, 726615, 729028, 746031, 748476, 753378, 755835, 760761, 768180, 775635, 780625, 798216, 831405, 844350, 849556, 852165, 854778, 860016, 865270, 875826, 881128, 883785, 886446, 894453, 899811, 905185, 918690, 937765, 946000, 957036, 973710, 979300, 982101, 996166, 1001820, 1010331, 1013176, 1021735, 1030330, 1038961, 1047628, 1059240, 1062153, 1065070, 1067991, 1082656, 1103355, 1109305, 1115271, 1121253, 1139295, 1160526, 1181953, 1185030, 1197378, 1203576, 1209790, 1222266, 1225395, 1234806, 1244253, 1256905, 1260078, 1263255, 1272810, 1276003, 1285606, 1288815, 1295245, 1304920, 1317876, 1327635, 1363726, 1373653, 1380291, 1390278, 1407003, 1430586, 1447551, 1481781, 1485226, 1512930, 1516411, 1530375, 1533876, 1537381, 1544403, 1547920, 1551441, 1562028, 1590436, 1597578, 1608321.

Some larger examples :