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centered octagonal n.  and  pernicious n.

First numbers that belong to both sets:

9, 25, 49, 81, 121, 289, 361, 529, 625, 841, 961, 1089, 1225, 1521, 1681, 1849, 2401, 2601, 3025, 3481, 4225, 4489, 5041, 5929, 6241, 7225, 7569, 8281, 9025, 9409, 10609, 12769, 14641, 15625, 16129, 16641, 17161, 18225, 21025, 21609, 24649, 32761, 34225, 35721, 37249, 43681, 46225, 47089, 49729, 50625, 51529, 53361, 55225, 66049, 67081, 69169, 77841, 88209, 94249, 95481, 97969, 101761, 121801, 123201, 127449, 131769, 139129, 140625, 143641, 146689, 148225, 160801, 164025, 165649, 178929, 180625, 184041, 198025, 201601, 208849, 218089, 219961, 223729, 229441, 235225, 237169, 243049, 251001, 263169, 265225, 269361, 290521, 294849, 297025, 299209, 312481, 319225, 321489, 326041, 330625, 339889, 354025, 375769, 383161, 405769, 423801, 436921, 442225, 444889, 450241, 458329, 466489, 474721, 480249, 485809, 488601, 491401, 511225, 516961, 525625, 528529, 543169, 555025, 591361, 619369, 625681, 638401, 644809, 648025, 654481, 664225, 690561, 720801, 734449, 755161, 758641, 765625, 776161, 801025, 804609, 822649, 826281, 840889, 848241, 866761, 881721, 904401, 908209, 931225, 938961, 962361, 970225, 974169, 982081, 990025, 1006009, 1014049, 1022121, 1030225, 1038361, 1050625, 1054729, 1062961, 1104601, 1113025, 1142761, 1172889, 1177225, 1181569, 1207801, 1216609, 1225449, 1243225, 1261129, 1265625, 1283689, 1306449, 1320201, 1352569, 1357225, 1361889, 1371241, 1380625, 1418481, 1428025, 1432809, 1437601, 1442401, 1452025, 1456849, 1461681, 1471369, 1495729, 1535121, 1555009, 1565001, 1575025, 1580049, 1666681, 1677025, 1692601, 1703025, 1734489, 1776889, 1782225, 1787569, 1792921, 1814409, 1819801, 1825201, 1836025, 1857769, 1874161, 1885129, 1912689, 1923769, 1929321, 1940449, 1957201, 1962801, 1974025, 1996569, 2002225, 2007889, 2024929, 2036329, 2047761, 2053489, 2070721, 2076481, 2082249, 2122849, 2163841, 2211169, 2223081, 2277081, 2283121, 2301289, 2325625, 2362369, 2418025, 2436721, 2442969, 2461761, 2480625, 2486929, 2499561, 2518569, 2531281, 2537649, 2544025, 2550409, 2569609, 2582449, 2608225, 2686321, 2719201, 2739025, 2745649, 2792241, 2798929, 2832489, 2845969, 2879809, 2907025, 2927521, 2954961, 2961841, 2968729, 2989441, 2996361, 3003289, 3059001, 3094081, 3101121, 3108169, 3150625, 3171961, 3179089, 3214849.

Some larger examples :