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Jordan-Polya n.  and  square n.

First numbers that belong to both sets:

1, 4, 16, 36, 64, 144, 256, 576, 1024, 1296, 2304, 4096, 5184, 9216, 14400, 16384, 20736, 36864, 46656, 57600, 65536, 82944, 147456, 186624, 230400, 262144, 331776, 518400, 589824, 746496, 921600, 1048576, 1327104, 1679616, 2073600, 2359296, 2985984, 3686400, 4194304, 5308416, 6718464, 8294400, 9437184, 11943936, 14745600, 16777216, 18662400, 21233664, 25401600, 26873856, 33177600, 37748736, 47775744, 58982400, 60466176, 67108864, 74649600, 84934656, 101606400, 107495424, 132710400, 150994944, 191102976, 207360000, 235929600, 241864704, 268435456, 298598400, 339738624, 406425600, 429981696, 530841600, 603979776, 671846400, 764411904, 829440000, 914457600, 943718400, 967458816, 1073741824, 1194393600, 1358954496, 1625702400, 1719926784, 2123366400, 2176782336, 2415919104, 2687385600, 3057647616, 3317760000, 3657830400, 3774873600, 3869835264, 4294967296, 4777574400, 5435817984, 6502809600, 6879707136, 7464960000, 8493465600, 8707129344, 9663676416, 10749542400, 12230590464, 13271040000, 14631321600, 15099494400, 15479341056, 17179869184, 19110297600, 21743271936, 24186470400, 26011238400, 27518828544, 29859840000, 32920473600, 33973862400, 34828517376, 38654705664, 42998169600, 48922361856, 53084160000, 58525286400, 60397977600, 61917364224, 68719476736, 76441190400, 78364164096, 86973087744, 96745881600, 104044953600, 110075314176, 119439360000, 131681894400, 135895449600, 139314069504, 154618822656, 171992678400, 195689447424, 212336640000, 234101145600, 241591910400, 247669456896, 268738560000, 274877906944, 305764761600, 313456656384, 347892350976, 365783040000, 386983526400, 416179814400, 440301256704, 477757440000, 526727577600, 543581798400, 557256278016, 618475290624, 687970713600, 782757789696, 849346560000, 870712934400, 936404582400, 966367641600, 990677827584, 1074954240000, 1099511627776, 1185137049600, 1223059046400, 1253826625536, 1391569403904, 1463132160000, 1547934105600, 1664719257600, 1761205026816, 1911029760000, 2106910310400, 2174327193600, 2229025112064, 2473901162496, 2751882854400, 2821109907456, 2985984000000, 3131031158784, 3397386240000, 3482851737600, 3745618329600, 3865470566400, 3962711310336, 4299816960000, 4398046511104, 4740548198400, 4892236185600, 5015306502144, 5566277615616, 5852528640000, 6191736422400, 6658877030400, 7044820107264, 7644119040000, 8427641241600, 8697308774400, 8916100448256, 9674588160000, 9895604649984, 11007531417600, 11284439629824, 11943936000000, 12524124635136, 13168189440000, 13589544960000, 13931406950400, 14982473318400, 15461882265600, 15850845241344, 17199267840000, 17592186044416, 18962192793600, 19568944742400, 20061226008576, 22265110462464, 23410114560000, 24766945689600, 26635508121600, 28179280429056, 30576476160000, 31345665638400, 33710564966400, 34789235097600, 35664401793024, 38698352640000, 39582418599936, 42664933785600, 44030125670400, 45137758519296, 47775744000000, 50096498540544, 52672757760000, 54358179840000, 55725627801600, 59929893273600, 61847529062400, 63403380965376, 68797071360000, 70368744177664, 75848771174400, 78275778969600, 80244904034304, 89060441849856, 93640458240000, 99067782758400, 101559956668416, 106542032486400, 107495424000000, 112717121716224, 122305904640000, 125382662553600, 134842259865600, 139156940390400, 142657607172096, 154793410560000, 158329674399744, 170659735142400, 176120502681600, 180551034077184, 191102976000000, 200385994162176, 210691031040000, 217432719360000, 222902511206400, 239719573094400, 247390116249600, 253613523861504, 275188285440000, 281474976710656, 303395084697600, 313103115878400, 320979616137216, 348285173760000, 356241767399424, 374561832960000.

Some larger examples :