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junction n.  and  square n.

First numbers that belong to both sets:

1024, 2025, 4624, 5625, 7225, 10609, 11025, 17424, 18225, 19321, 21316, 21609, 23409, 23716, 24025, 27225, 28224, 30625, 34225, 35721, 38025, 41209, 42025, 44521, 47524, 50625, 53824, 57121, 60025, 61009, 65025, 68121, 70225, 71824, 81225, 87025, 93025, 114921, 119025, 120409, 130321, 133225, 140625, 148225, 156025, 163216, 164025, 178929, 180625, 192721, 205209, 216225, 219024, 225625, 232324, 239121, 245025, 246016, 255025, 256036, 265225, 273529, 277729, 283024, 308025, 314721, 319225, 332929, 342225, 354025, 366025, 403225, 408321, 416025, 429025, 446224, 465124, 469225, 483025, 505521, 511225, 515524, 540225, 555025, 570025, 616225, 622521, 632025, 669124, 677329, 703921, 714025, 731025, 741321, 748225, 769129, 788544, 790321, 851929, 859329, 874225, 879844, 912025, 913936, 923521, 931225, 954529, 964324, 988036, 1006009, 1010025, 1054729, 1065024, 1071225, 1092025, 1110916, 1113025, 1134225, 1140624, 1159929, 1170724, 1201216, 1221025, 1234321, 1281424, 1288225, 1311025, 1334025, 1364224, 1375929, 1390041, 1404225, 1428025, 1430416, 1452025, 1468944, 1517824, 1525225, 1535121, 1547536, 1600225, 1607824, 1630729, 1651225, 1661521, 1674436, 1677025, 1700416, 1703025, 1729225, 1760929, 1852321, 1855044, 1879641, 1885129, 1896129, 1918225, 1974025, 2010724, 2024929, 2030625, 2111209, 2117025, 2119936, 2134521, 2146225, 2155024, 2169729, 2175625, 2181529, 2205225, 2217121, 2235025, 2241009, 2262016, 2265025, 2304324, 2347024, 2418025, 2449225, 2468041, 2486929, 2502724, 2544025, 2683044, 2706025, 2739025, 2805625, 2819041, 2829124, 2951524, 3045025, 3150625, 3157729, 3196944, 3305124, 3330625, 3337929, 3367225, 3404025, 3441025, 3444736, 3463321, 3508129, 3541924, 3632836, 3705625, 3713329, 3755844, 3783025, 3822025, 3849444, 3884841, 3892729, 3908529, 3976036, 4012009, 4020025, 4044121, 4072324, 4100625, 4129024, 4177936, 4182025, 4223025, 4297329, 4359744, 4431025, 4498641, 4507129, 4515625, 4545424, 4601025, 4721929, 4813636, 4818025, 4866436, 4892944, 4906225, 4941729, 4968441, 5040025, 5130225, 5166529, 5184729, 5313025, 5317636, 5340721, 5414929, 5438224, 5550736, 5607424, 5640625, 5650129, 5659641, 5736025, 5788836, 5817744, 5870929, 5890329, 5900041, 6027025, 6076225, 6091024, 6160324, 6175225, 6220036, 6225025, 6305121, 6340324, 6365529, 6385729, 6395841, 6411024, 6528025, 6563844, 6610041, 6620329, 6630625, 6640929.

Some larger examples :