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gapful n.  and  hexagonal n.

First numbers that belong to both sets:

120, 190, 231, 561, 1035, 1540, 1770, 3003, 4005, 4186, 4560, 5151, 5995, 7140, 7875, 9180, 11175, 11781, 12090, 12720, 13695, 15400, 16110, 17205, 17955, 18336, 19110, 19900, 23220, 25425, 28680, 30135, 30628, 32640, 35245, 37950, 39060, 41328, 46665, 50403, 54285, 61776, 64620, 66066, 79800, 86320, 97461, 101025, 103740, 105570, 106491, 108345, 109278, 112101, 113050, 114960, 117855, 119805, 122760, 124750, 129795, 130816, 132870, 134940, 138075, 143380, 145530, 147696, 148785, 150975, 154290, 156520, 157641, 161028, 162165, 165600, 167910, 171405, 176121, 177310, 178503, 179700, 183315, 185745, 189420, 191890, 198135, 204480, 210925, 214840, 220116, 230860, 240471, 241860, 254541, 258840, 263175, 264628, 270480, 274911, 280875, 288420, 303810, 306936, 313236, 316410, 324415, 332520, 335790, 340725, 352380, 365940, 367653, 372816, 383250, 386760, 395605, 400960, 406351, 413595, 430128, 445096, 446985, 460320, 475800, 489555, 546535, 554931, 574056, 578350, 593505, 600060, 611065, 626640, 653796, 683865, 690900, 768180, 778128, 793170, 818560, 857395, 878475, 902496, 910575, 924120, 973710, 987715, 1001820, 1004653, 1007490, 1016025, 1030330, 1036080, 1038961, 1044735, 1050525, 1059240, 1065070, 1076778, 1079715, 1082656, 1085601, 1088550, 1094460, 1103355, 1118260, 1124250, 1130256, 1133265, 1139295, 1148370, 1154440, 1169685, 1178880, 1185030, 1194285, 1197378, 1209790, 1216020, 1219141, 1225395, 1231665, 1237951, 1241100, 1247410, 1263255, 1272810, 1279200, 1288815, 1304920, 1311390, 1321125, 1327635, 1330896, 1337430, 1340703, 1343980, 1357128, 1360425, 1370340, 1376970, 1380291, 1383616, 1386945, 1403650, 1407003, 1410360, 1420455, 1427205, 1433971, 1437360, 1444150, 1461195, 1471470, 1478340, 1488675, 1492128, 1505980, 1512930, 1523385, 1530375, 1540890, 1547920, 1565565, 1576200, 1583310, 1586871, 1594005, 1604736, 1608321, 1611910, 1619100, 1629915, 1637145, 1648020, 1655290, 1662576, 1669878, 1673535, 1684530, 1691880, 1702935, 1721440, 1728870, 1740045, 1747515, 1758750, 1766260, 1770021, 1785105, 1796460, 1800253, 1804050, 1815465, 1819278, 1830741, 1834570, 1842240, 1853775, 1861485, 1873080, 1876953, 1880830, 1900275, 1904176, 1911990, 1919820, 1931595, 1951300, 1959210, 1967136, 1971105, 1979055, 1991010, 1999000, 2003001, 2031120, 2051325, 2079780.

Some larger examples :