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heptagonal n.  and  pernicious n.

First numbers that belong to both sets:

7, 18, 34, 55, 81, 112, 148, 286, 342, 403, 1177, 1288, 1404, 1651, 2356, 2512, 2839, 3367, 3553, 3744, 3940, 4141, 4558, 5929, 6175, 7480, 7756, 8910, 9517, 10791, 12496, 13579, 14326, 16687, 17935, 19228, 21022, 22896, 24850, 25351, 25857, 26368, 26884, 28998, 31753, 33466, 34633, 36421, 38875, 40132, 40768, 42055, 43362, 46717, 49491, 51624, 53071, 53802, 54538, 55279, 60606, 62173, 62964, 63760, 65367, 66994, 68641, 69472, 73702, 74563, 79834, 81631, 82537, 84364, 86211, 88078, 90916, 93799, 95746, 97713, 98704, 107848, 114169, 118483, 128482, 130759, 131905, 134212, 135373, 140067, 147258, 152152, 154629, 166023, 168610, 177822, 180499, 191407, 201214, 208369, 209815, 217120, 221563, 223054, 227557, 238239, 241336, 246019, 249166, 250747, 253924, 266832, 268468, 285103, 290191, 295324, 297045, 307476, 309232, 310993, 312759, 319873, 325261, 327067, 334341, 336172, 338008, 341695, 343546, 362331, 364237, 371911, 385533, 391446, 393427, 395413, 397404, 401401, 405418, 415548, 419635, 423742, 425803, 427869, 429940, 436183, 450925, 453051, 470239, 472410, 474586, 483340, 489958, 496621, 514609, 542191, 546858, 556252, 558613, 570493, 572884, 577681, 580087, 587335, 592192, 594628, 599515, 611820, 614296, 634284, 636805, 644398, 646939, 659719, 664866, 677821, 680427, 685654, 688275, 696168, 698809, 714760, 722803, 725494, 733597, 744471, 749938, 769230, 777573, 785961, 791578, 817102, 819963, 828576, 848848, 851764, 860542, 866419, 872316, 878233, 884170, 890127, 905107, 908118, 911134, 944640, 953883, 956974, 963171, 969388, 1003939, 1007110, 1010286, 1016653, 1029447, 1032658, 1042321, 1045552, 1048788, 1078137, 1081423, 1088010, 1094617, 1101244, 1104565, 1111222, 1127952, 1148193, 1151584, 1154980, 1165198, 1168614, 1175461, 1178892, 1185769, 1189215, 1196122, 1199583, 1203049, 1213477, 1227451, 1255639, 1259185, 1262736, 1276990, 1287733, 1291324, 1305738, 1309354, 1316601, 1320232, 1338462, 1345789, 1360503, 1371591, 1379008, 1393902, 1416393, 1435273, 1442860, 1481095, 1492663, 1500400, 1508157, 1512043, 1515934, 1519830, 1527637, 1531548, 1535464, 1559065, 1563016, 1574899, 1578870, 1602801, 1610818, 1614834, 1667497, 1671583, 1683871.

Some larger examples :