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junction n.  and  powerful n.

First numbers that belong to both sets:

216, 1024, 2025, 4624, 4913, 5324, 5625, 7225, 10609, 11025, 13824, 14112, 16928, 17424, 18225, 19321, 21316, 21609, 23409, 23716, 24025, 27225, 28125, 28224, 30625, 31212, 34225, 35721, 38025, 41209, 42025, 44521, 47524, 49928, 50625, 53824, 55125, 57121, 59319, 60025, 61009, 62424, 65025, 65219, 66125, 68121, 70225, 71824, 78732, 81225, 87025, 91125, 93025, 95832, 99127, 114921, 119025, 120409, 122825, 123008, 130321, 133225, 136125, 140625, 148225, 156025, 163216, 164025, 178929, 180625, 190125, 192721, 202612, 205209, 216225, 219024, 223112, 225625, 231525, 232324, 233523, 235224, 238328, 239121, 239432, 245025, 246016, 248832, 253125, 255025, 256036, 256328, 265225, 265837, 268119, 273529, 274625, 277729, 283024, 292032, 308025, 309123, 314721, 319225, 325125, 332929, 342225, 354025, 366025, 378125, 381024, 390224, 403225, 406125, 408321, 416025, 429025, 435125, 446224, 463736, 465124, 469225, 483025, 492032, 493039, 505521, 508032, 511225, 515524, 540225, 552123, 555025, 561125, 570025, 575532, 596232, 599427, 613832, 616225, 622521, 632025, 666125, 669124, 674028, 677329, 700928, 703125, 703921, 714025, 729632, 731025, 739328, 741125, 741321, 748225, 769129, 788544, 790321, 798848, 820125, 821516, 851929, 855432, 859329, 865928, 874225, 879844, 903125, 912025, 913936, 923521, 931225, 954529, 957728, 964324, 985527, 988036, 990125, 995328, 1005723, 1006009, 1010025, 1054729, 1065024, 1071225, 1081125, 1092025, 1092727, 1105425, 1110916, 1113025, 1128125, 1131008, 1134225, 1140624, 1149128, 1159929, 1170724, 1191016, 1201216, 1221025, 1225125, 1234321, 1257728, 1267232, 1281424, 1288225, 1292832, 1293732, 1295029, 1305728, 1311025, 1326125, 1334025, 1364224, 1367631, 1373125, 1375929, 1390041, 1404225, 1404928, 1417928, 1428025, 1430416, 1452025, 1458632, 1468944, 1485125, 1517824, 1525225, 1535121, 1539727, 1540125, 1547536, 1586032, 1600225, 1605632, 1607824, 1630729, 1633023, 1633932, 1634432, 1648928, 1651225, 1661521, 1674436, 1677025, 1700416, 1703025, 1711125, 1723025, 1729225, 1741932, 1760929, 1774728, 1797228, 1826132, 1830125, 1835528, 1852321, 1855044, 1879641, 1885129, 1896129, 1906624, 1918225, 1953125, 1974025, 1984032, 2010724, 2016032, 2024929, 2030625, 2056223, 2080125, 2111209, 2117025, 2119936, 2134521, 2145125.

Some larger examples :