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centered square n.  and  sphenic n.

First numbers that belong to both sets:

1105, 2665, 3445, 7565, 8845, 14965, 15665, 16745, 17485, 18241, 20605, 22685, 23545, 27145, 28085, 32005, 32513, 35113, 37265, 48985, 50245, 50881, 55445, 56785, 62305, 71065, 74885, 78013, 80401, 81205, 84461, 85285, 88621, 89465, 109045, 111865, 113765, 116645, 118585, 119561, 123505, 124501, 136765, 152905, 158485, 159613, 160745, 169945, 178205, 179401, 181805, 187885, 192821, 194065, 200345, 209305, 214513, 217141, 219785, 222445, 235985, 245701, 247105, 254185, 257045, 261365, 264265, 268645, 271585, 279005, 294145, 297221, 306545, 308113, 309685, 316013, 322405, 328861, 330485, 341965, 379321, 393385, 395161, 402305, 407705, 414961, 422281, 446513, 465613, 467545, 473365, 477265, 483145, 487085, 499001, 503005, 513085, 517145, 523265, 527365, 533545, 537685, 546013, 562861, 573521, 577813, 579965, 586445, 601705, 606101, 614941, 635065, 641845, 648661, 650941, 664705, 699745, 709241, 721201, 723605, 740545, 747865, 765085, 767561, 797585, 802645, 805181, 815365, 817921, 823045, 828185, 838513, 841105, 846301, 851513, 861985, 867245, 872521, 875165, 880465, 885781, 888445, 899141, 915305, 928885, 945313, 956345, 959113, 984205, 992641, 1018165, 1032485, 1038241, 1049801, 1055605, 1067261, 1070185, 1087813, 1129505, 1144585, 1150645, 1159765, 1162813, 1181185, 1196605, 1199701, 1205905, 1243465, 1252945, 1256113, 1284805, 1300885, 1307345, 1317065, 1323565, 1339885, 1366205, 1376141, 1396121, 1399465, 1416245, 1422985, 1439905, 1443301, 1463761, 1477481, 1480921, 1487813, 1491265, 1501645, 1508585, 1540013, 1554085, 1571765, 1578865, 1582421, 1585981, 1589545, 1596685, 1643485, 1654381, 1665313, 1679945, 1694641, 1709401, 1716805, 1735385, 1742845, 1754065, 1769081, 1772845, 1799305, 1829785, 1837445, 1852813, 1864381, 1875985, 1907105, 1918841, 1926685, 1934545, 1954265, 1962181, 1966145, 1994005, 2034145, 2038181, 2046265, 2054365, 2066545, 2074685, 2082841, 2095105, 2099201, 2107405, 2127985, 2144521, 2156965, 2173613, 2177785, 2198705, 2211305, 2215513, 2228161, 2253565, 2266321, 2283385, 2304805, 2322013, 2369665, 2382745, 2391485, 2400241, 2413405, 2426605, 2431013, 2448685, 2457545, 2488681, 2493145, 2511041, 2520013, 2538005, 2560585, 2596921, 2606045, 2615185, 2638105, 2651905, 2674985, 2688881, 2702813, 2707465, 2712121, 2768305, 2787161, 2791885, 2801345, 2806081, 2839345.

Some larger examples :