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octagonal n.  and  pernicious n.

First numbers that belong to both sets:

21, 40, 65, 96, 133, 176, 280, 341, 481, 560, 936, 1160, 1281, 1408, 1825, 1976, 2133, 2465, 2821, 3008, 3400, 3816, 4033, 4256, 4485, 4720, 5208, 5461, 5985, 6256, 6816, 7105, 7400, 7701, 8008, 8321, 8965, 10680, 11408, 12936, 13736, 14145, 14981, 17176, 18565, 19040, 19521, 20501, 22533, 23056, 25208, 26320, 28033, 29205, 29800, 34133, 36080, 39445, 40136, 43681, 45141, 46625, 48133, 48896, 50440, 52008, 52801, 54405, 58520, 68101, 69921, 70840, 71765, 75525, 77441, 82336, 83333, 84336, 86360, 90480, 92576, 99008, 100101, 101200, 107920, 109061, 110208, 111361, 116033, 120801, 126896, 131880, 133141, 134408, 136960, 143445, 150080, 152776, 156865, 172320, 182533, 193040, 199176, 202280, 203841, 206981, 218160, 221408, 229633, 231296, 234640, 249985, 253461, 256961, 260485, 264033, 265816, 269400, 271201, 273008, 274821, 283976, 285825, 287680, 291408, 295160, 297045, 304645, 306560, 308481, 312341, 328021, 332001, 334000, 342056, 346120, 348161, 350208, 356385, 360533, 364705, 368901, 371008, 405536, 409960, 425633, 434721, 443905, 457861, 464920, 472033, 479200, 491265, 505941, 508408, 510881, 523336, 530881, 559008, 561601, 590520, 593185, 614721, 636641, 656136, 661760, 678776, 690240, 704705, 707616, 710533, 716385, 731120, 749000, 755008, 767096, 791560, 797736, 800833, 807045, 835296, 841640, 860816, 873720, 883461, 896533, 903105, 916321, 919640, 929633, 966736, 970145, 973560, 976981, 980408, 987280, 1032533, 1039585, 1050208, 1053761, 1057320, 1064456, 1068033, 1071616, 1078800, 1082401, 1086008, 1089621, 1115080, 1118741, 1144536, 1174376, 1178133, 1231361, 1242920, 1246785, 1285765, 1289696, 1293633, 1301525, 1305480, 1309441, 1313408, 1317381, 1321360, 1341345, 1369576, 1373633, 1377696, 1398101, 1406305, 1422785, 1426920, 1431061, 1439361, 1443520, 1456033, 1498133, 1502376, 1515141, 1519408, 1532245, 1545136, 1553760, 1566741, 1571080, 1575425, 1584133, 1597240, 1623616, 1636885, 1677016, 1690501, 1735841, 1740408, 1772545, 1800325, 1818965, 1828321, 1833008, 1894485, 1908816, 1928008, 1932821, 1942465, 1947296, 1952133, 1971541, 1986160, 2000833, 2030341, 2045176, 2050133, 2060065, 2070021, 2110085, 2130261, 2135320, 2150533, 2155616.

Some larger examples :