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congruent n.  and  powerful n.

First numbers that belong to both sets:

125, 216, 343, 500, 864, 1125, 1372, 1944, 2000, 2197, 2744, 3087, 3125, 3375, 3456, 4500, 5400, 5488, 6125, 7776, 8000, 8575, 8788, 9261, 10125, 10584, 10648, 10976, 12167, 12348, 12500, 13500, 13824, 15125, 16807, 17496, 18000, 19773, 21125, 21600, 21952, 24389, 24500, 24696, 26136, 27000, 27783, 28125, 29791, 30375, 31104, 32000, 34300, 35152, 36125, 36504, 37044, 39304, 40500, 41503, 42336, 42592, 43904, 45125, 48600, 48668, 49392, 50000, 50653, 54000, 54872, 54925, 55125, 55296, 57967, 59319, 60500, 62424, 66125, 67228, 68600, 68921, 69984, 72000, 77175, 77976, 78125, 79092, 83349, 84375, 84500, 86400, 87808, 91125, 95256, 95832, 97336, 97556, 98000, 98784, 99127, 103823, 104544, 105125, 107653, 108000, 109503, 111132, 112500, 114264, 119164, 120125, 121500, 123823, 124416, 128000, 134456, 135000, 136125, 137200, 140608, 144500, 146016, 148176, 148877, 151263, 153125, 157216, 157464, 162000, 165375, 166012, 166375, 169344, 170368, 171125, 175616, 177957, 180500, 181447, 181656, 190125, 194400, 194672, 197568, 200000, 202612, 207576, 210125, 214375, 216000, 219488, 219501, 219700, 220500, 221184, 222264, 226981, 231125, 231525, 231868, 235224, 237276, 238328, 242000, 243000, 249696, 250047, 253125, 264500, 264600, 265837, 266200, 268119, 268912, 273375, 274400, 274625, 275684, 276125, 279936, 288000, 288463, 295704, 300125, 304175, 308700, 311904, 312500, 316368, 325125, 328509, 328536, 329623, 332024, 333396, 337500, 338000, 343000, 345600, 351125, 351232, 353736, 357911, 363096, 364500, 371293, 373527, 378125, 381024, 383328, 389344, 390224, 392000, 395136, 396508, 399384, 406125, 408375, 415292, 418176, 420175, 420500, 430612, 432000, 435125, 437400, 438012, 444528, 450000, 453789, 455877, 456533, 457056, 463736, 465125, 469567, 474552, 476656, 477144, 480500, 486000, 493039, 493848, 494325, 495292, 496125, 497664, 512000, 518616, 521703, 521752, 528125, 533871, 537824, 540000, 544500, 548800, 561125, 561816, 562432, 570375, 576583, 578000, 584064, 592704, 595125, 595508, 596183, 605052, 606744, 609725, 612500, 614125, 617400, 620289, 628864, 629856, 630125, 634207.

Some larger examples :