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centered square n.  and  odious n.

First numbers that belong to both sets:

1, 13, 25, 41, 61, 145, 181, 265, 313, 421, 481, 545, 613, 761, 841, 925, 1201, 1301, 1513, 1741, 2113, 2245, 2521, 2813, 2965, 3121, 3613, 3785, 3961, 4141, 4513, 4705, 5101, 5305, 6385, 6845, 7321, 7813, 8065, 8321, 8581, 9113, 9661, 10225, 10513, 10805, 11101, 11705, 12325, 13285, 13945, 14281, 14965, 15313, 16021, 16381, 17113, 17861, 18625, 19405, 20201, 21841, 23113, 23545, 24865, 25313, 25765, 26221, 26681, 27613, 28561, 29525, 30505, 31501, 33025, 33541, 34585, 35645, 36721, 37813, 38921, 40045, 40613, 44105, 45301, 46513, 47125, 47741, 48985, 50881, 52165, 52813, 54121, 56113, 58141, 58825, 59513, 60901, 61601, 62305, 63013, 63725, 65885, 69565, 70313, 71821, 73345, 74113, 75661, 77225, 78013, 80401, 82013, 82825, 83641, 88621, 89465, 90313, 92021, 92881, 93745, 94613, 96361, 99013, 100801, 104425, 109045, 109981, 111865, 114721, 115681, 117613, 118585, 121525, 123505, 125501, 127513, 129541, 130561, 131585, 132613, 134681, 136765, 138865, 140981, 144185, 145261, 147425, 148513, 149605, 150701, 151801, 156241, 157361, 158485, 159613, 160745, 161881, 163021, 165313, 167621, 168781, 169945, 171113, 173461, 177013, 178205, 180601, 181805, 186661, 187885, 191581, 194065, 197821, 200345, 201613, 202885, 206725, 208013, 209305, 211901, 218461, 219785, 221113, 222445, 223781, 225121, 226465, 229165, 230521, 231881, 233245, 237361, 238741, 240125, 242905, 244301, 245701, 248513, 255613, 258481, 262813, 264265, 267181, 271585, 276025, 277513, 279005, 283505, 285013, 289561, 292613, 295681, 298765, 301865, 303421, 308113, 309685, 312841, 314425, 317605, 319201, 322405, 324013, 327241, 332113, 335381, 340313, 345281, 348613, 350285, 360401, 367225, 368941, 377581, 379321, 382813, 386321, 388081, 395161, 396941, 400513, 402305, 405901, 407705, 411325, 413141, 420445, 422281, 424121, 425965, 427813, 429665, 431521, 433381, 435245, 440861, 444625, 450301, 452201, 454105, 461761, 463685, 465613, 467545, 469481, 481181, 485113, 487085, 491041, 495013, 501001, 503005, 507025, 511061, 515113, 519181, 525313, 527365, 531481.

Some larger examples :