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centered square n.  and  pernicious n.

First numbers that belong to both sets:

5, 13, 25, 41, 61, 145, 181, 265, 313, 421, 481, 545, 613, 761, 841, 925, 1201, 1301, 1513, 1741, 2113, 2245, 2521, 2965, 3121, 3613, 3785, 4141, 4513, 4705, 5305, 6385, 7321, 7813, 8065, 8321, 8581, 9113, 10513, 10805, 12325, 16381, 17113, 17861, 18625, 21841, 23113, 23545, 24865, 25313, 25765, 26681, 27613, 33025, 33541, 34585, 38921, 44105, 47125, 47741, 48985, 50881, 60901, 61601, 63725, 65885, 69565, 70313, 71821, 73345, 74113, 80401, 82013, 82825, 89465, 90313, 92021, 99013, 100801, 104425, 109045, 109981, 111865, 114721, 117613, 118585, 121525, 125501, 131585, 132613, 134681, 145261, 147425, 148513, 149605, 156241, 159613, 160745, 163021, 165313, 169945, 177013, 187885, 191581, 202885, 211901, 218461, 221113, 222445, 225121, 229165, 233245, 237361, 240125, 242905, 244301, 245701, 255613, 258481, 262813, 264265, 271585, 277513, 295681, 309685, 312841, 319201, 322405, 324013, 327241, 332113, 345281, 360401, 367225, 377581, 379321, 382813, 388081, 400513, 402305, 411325, 413141, 420445, 424121, 433381, 440861, 452201, 454105, 465613, 469481, 481181, 485113, 487085, 491041, 495013, 503005, 507025, 511061, 515113, 519181, 525313, 527365, 531481, 552301, 556513, 571381, 586445, 588613, 590785, 603901, 608305, 612725, 621613, 630565, 632813, 641845, 653225, 660101, 676285, 678613, 680945, 685621, 690313, 709241, 714013, 716405, 718801, 721201, 726013, 728425, 730841, 735685, 747865, 767561, 777505, 782501, 787513, 790025, 833341, 838513, 846301, 851513, 867245, 888445, 891113, 893785, 896461, 907205, 909901, 912601, 918013, 928885, 937081, 942565, 956345, 961885, 964661, 970225, 978601, 981401, 987013, 998285, 1001113, 1003945, 1012465, 1018165, 1023881, 1026745, 1035361, 1038241, 1041125, 1061425, 1081921, 1105585, 1111541, 1138541, 1141561, 1150645, 1162813, 1181185, 1209013, 1218361, 1221485, 1230881, 1240313, 1243465, 1249781, 1259285, 1265641, 1268825, 1272013, 1275205, 1284805, 1291225, 1304113, 1343161, 1359601, 1369513, 1372825, 1396121, 1399465, 1416245, 1422985, 1439905, 1453513, 1463761, 1477481, 1480921, 1484365, 1494721, 1498181, 1501645, 1529501, 1547041, 1550561, 1554085, 1575313, 1585981, 1589545, 1607425.

Some larger examples :