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centered square n.  and  evil n.

First numbers that belong to both sets:

5, 85, 113, 221, 365, 685, 1013, 1105, 1405, 1625, 1861, 1985, 2381, 2665, 3281, 3445, 4325, 4901, 5513, 5725, 5941, 6161, 6613, 7081, 7565, 8845, 9385, 9941, 11401, 12013, 12641, 12961, 13613, 14621, 15665, 16745, 17485, 18241, 19013, 19801, 20605, 21013, 21425, 22261, 22685, 23981, 24421, 27145, 28085, 29041, 30013, 31001, 32005, 32513, 34061, 35113, 36181, 37265, 38365, 39481, 41185, 41761, 42341, 42925, 43513, 44701, 45905, 48361, 49613, 50245, 51521, 53465, 54781, 55445, 56785, 57461, 60205, 64441, 65161, 66613, 67345, 68081, 68821, 71065, 72581, 74885, 76441, 78805, 79601, 81205, 84461, 85285, 86113, 86945, 87781, 91165, 95485, 97241, 98125, 99905, 101701, 102605, 103513, 105341, 106261, 107185, 108113, 110921, 112813, 113765, 116645, 119561, 120541, 122513, 124501, 126505, 128525, 133645, 135721, 137813, 139921, 142045, 143113, 146341, 152905, 154013, 155125, 164165, 166465, 172285, 174641, 175825, 179401, 183013, 184225, 185441, 189113, 190345, 192821, 195313, 196565, 199081, 204161, 205441, 210601, 213205, 214513, 215825, 217141, 227813, 234613, 235985, 241513, 247105, 249925, 251341, 252761, 254185, 257045, 259921, 261365, 265721, 268645, 270113, 273061, 274541, 280501, 282001, 286525, 288041, 291085, 294145, 297221, 300313, 304981, 306545, 311261, 316013, 320801, 325625, 328861, 330485, 333745, 337021, 338665, 341965, 343621, 346945, 351961, 353641, 355325, 357013, 358705, 362101, 363805, 365513, 370661, 372385, 374113, 375845, 381065, 384565, 389845, 391613, 393385, 398725, 404101, 409513, 414961, 416785, 418613, 437113, 438985, 442741, 446513, 448405, 456013, 457925, 459841, 471421, 473365, 475313, 477265, 479221, 483145, 489061, 493025, 497005, 499001, 505013, 509041, 513085, 517145, 521221, 523265, 529421, 533545, 537685, 541841, 546013, 548105, 554405, 558625, 562861, 564985, 569245, 573521, 575665, 577813, 579965, 582121, 584281, 592961, 595141, 597325, 599513, 601705, 606101, 610513, 617161, 619385, 635065, 637321, 639581, 644113, 646385, 648661, 650941, 655513, 657805, 662401, 664705, 669325, 671641, 673961, 687965, 692665, 695021, 697381, 699745.

Some larger examples :