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centered square n.  and  gapful n.

First numbers that belong to both sets:

4141, 6161, 21425, 29525, 132613, 137813, 206725, 215825, 240125, 249925, 276025, 286525, 414961, 601705, 614941, 619385, 630565, 714013, 861985, 1015313, 1029613, 1209013, 1224613, 1419613, 1436513, 1647113, 1665313, 1891513, 1911013, 2014025, 2086925, 2115625, 2190325, 2219725, 2296225, 2326325, 2404625, 2435425, 2515525, 2547025, 2628925, 2661125, 2744825, 2777725, 2863225, 2896825, 2984125, 4092661, 4144321, 4383761, 5831113, 6297701, 6326125, 6361745, 6418945, 6454825, 6657601, 6738121, 6796985, 6833905, 6893185, 6930365, 7663613, 8132545, 8270245, 8376325, 8516065, 8832605, 8976085, 10328513, 10374013, 10927813, 10974613, 11544013, 11592113, 12177113, 12226513, 12827113, 12877813, 13494013, 13546013, 14177813, 14231113, 14878513, 14933113, 15596113, 15652013, 16330613, 16387813, 17082013, 17140513, 17850313, 17910113, 18635513, 18696613, 19437613, 19500013, 20116825, 20205725, 20435225, 20524825, 20756125, 20846425, 21079525, 21170525, 21405425, 21497125, 21733825, 21826225, 22064725, 22157825, 22398125, 22491925, 22734025, 22828525, 23072425, 23167625, 23413325, 23509225, 23756725, 23853325, 24102625, 24199925, 24451025, 24549025, 24801925, 24900625, 25155325, 25254725, 25511225, 25611325, 25869625, 25970425, 26230525, 26332025, 26593925, 26696125, 26959825, 27062725, 27328225, 27431825, 27699125, 27803425, 28072525, 28177525, 28448425, 28554125, 28826825, 28933225, 29207725, 29314825, 29591125, 29698925, 29977025, 40599061, 40761421, 41505161, 43608461, 44377621, 44547361, 45324721, 47521501, 48324281, 48501401, 49312381, 53096513, 53716613, 58698613, 59350513, 60137545, 60159481, 60247265, 60401041, 60423025, 60533005, 61505141, 61571705, 61682725, 61860565, 61971845, 63022765, 63135085, 63315005, 63427585, 64490725, 64604345, 64786345, 64900225, 65883721, 65975585, 66090505, 66274585, 66389765, 67036621, 67291601, 67477345, 67593565, 67779725, 67896205, 68456701, 68996005, 69113525, 69301765, 69419545, 71341513, 73750513, 80683105, 81115585, 81447085, 81881605, 82857065, 83295325, 83631245, 84071545, 85059925, 85503965, 85844305, 86290385, 87291685, 87741505, 88086265, 88538125, 89552345, 100323613, 100465313, 102173513, 102316513, 104040313, 104184613, 105924013, 106069613, 107824613, 107971513, 109742113, 109890313, 111676513, 111826013, 113627813, 113778613, 115596013, 115748113, 117581113, 117734513, 119583113, 119737813, 121602013, 121758013, 123637813, 123795113, 125690513, 125849113, 127760113, 127920013, 129846613, 130007813, 131950013, 132112513, 134070313, 134234113, 136207513, 136372613, 138361613, 138528013, 140532613, 140700313, 142720513, 142889513, 144925313, 145095613, 147147013, 147318613, 149385613, 149558513, 151641113, 151815313, 153913513, 154089013, 156202813, 156379613, 158509013.

Some larger examples :