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frugal n.  and  junction n.

First numbers that belong to both sets:

1024, 1215, 4913, 10112, 10609, 11008, 12005, 13824, 14823, 15123, 15309, 15616, 18225, 19321, 20923, 21609, 22016, 24010, 25625, 28125, 30625, 33614, 35721, 36015, 43011, 44521, 45619, 45625, 50421, 50625, 53125, 57121, 60025, 63423, 65625, 69629, 74431, 78732, 80919, 81920, 84035, 90625, 91125, 102016, 102625, 102912, 104615, 106112, 106625, 107125, 108032, 108621, 111005, 111104, 112023, 112509, 113024, 113627, 113920, 115625, 116125, 116224, 117625, 118016, 118125, 119021, 119936, 120320, 120409, 120625, 121014, 121216, 122825, 123008, 124625, 124928, 125824, 127616, 128625, 129033, 130321, 130816, 131712, 131726, 132023, 132921, 133225, 133427, 135625, 137728, 140211, 140625, 141426, 141824, 143125, 143717, 144722, 148736, 149931, 150016, 153819, 155520, 155625, 156025, 156735, 157221, 157922, 158123, 160623, 160625, 164025, 165632, 165731, 170723, 171520, 172529, 175625, 178119, 178125, 178929, 179334, 180625, 181521, 183125, 192721, 193027, 195223, 195625, 195938, 196830, 198125, 199439, 202612, 205312, 205625, 209223, 210625, 212625, 213109, 213111, 214326, 214528, 220323, 225625, 226719, 226723, 228125, 229635, 231822, 232211, 233125, 233216, 233523, 234009, 236032, 237824, 238328, 240818, 246016, 248125, 248832, 249318, 250112, 250625, 253125, 254421, 261023, 265625, 268119, 270625, 271313, 273529, 274625, 278528, 281219, 282123, 283318, 285625, 288125, 290625, 292922, 293125, 293327, 294637, 295323, 295424, 296227, 300811, 302526, 303616, 309729, 315217, 315625, 318125, 318719, 320031, 324821, 327321, 329216, 332929, 338125, 338432, 340625, 347733, 348823, 351122, 355023, 355625, 360625, 363125, 363527, 365117, 366025, 368128, 368219, 371522, 372224, 372519, 373827, 375625, 376832, 378125, 380023, 383125, 385024, 389632, 390224, 393728, 408125, 408321, 413125, 428125, 435125, 436736, 448335, 453125, 465625, 465831, 466432, 470528, 472827, 473125, 483328, 487936, 489845, 490625, 492032, 493039, 493533, 500823, 503125, 505521, 506530, 511413, 515625, 518125, 518319, 535423, 535625, 538731, 540625, 541717, 544231, 545027, 551124, 552727, 559433, 560623, 561125, 566029, 566433, 569037, 573839, 578125, 580625, 581632, 583929, 591219, 595625, 601526.

Some larger examples :