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Harshad n.  and  perfect power

First numbers that belong to both sets:

1, 4, 8, 9, 27, 36, 81, 100, 144, 216, 225, 243, 324, 400, 441, 512, 576, 900, 1296, 1521, 1728, 1764, 2025, 2304, 2401, 2601, 2704, 2916, 3600, 3969, 4225, 4356, 4624, 4913, 5184, 5832, 6084, 6400, 7056, 7744, 7776, 8000, 8100, 9216, 10000, 10404, 10816, 11025, 11664, 12100, 12321, 12544, 13689, 13824, 14400, 15876, 17424, 17576, 19044, 19600, 19683, 20736, 21904, 22500, 24336, 26244, 27000, 28224, 30276, 32400, 33124, 35937, 36100, 39204, 40000, 40401, 41616, 42849, 44100, 46656, 48400, 54756, 57600, 59049, 59319, 60516, 61504, 62001, 63504, 64000, 69696, 72900, 74088, 75625, 77841, 82944, 88209, 90000, 93636, 99225, 100000, 101124, 103041, 104329, 104976, 108900, 110592, 110889, 116964, 121104, 122500, 123201, 125000, 125316, 129600, 139876, 142884, 152100, 156816, 157464, 161604, 164836, 171396, 176400, 177147, 185193, 186624, 188356, 194481, 198025, 202500, 211600, 213444, 216000, 219024, 226576, 227529, 230400, 234256, 236196, 240100, 242064, 244036, 248004, 248832, 251001, 254016, 256036, 260100, 263169, 270400, 272484, 278784, 279936, 281961, 283024, 286225, 287496, 291600, 295936, 297025, 301401, 302500, 304704, 311364, 314432, 321489, 324900, 328509, 331776, 336400, 343000, 352836, 360000, 363609, 373248, 374544, 385641, 389376, 390625, 396900, 412164, 419904, 421875, 422500, 425104, 431649, 432964, 435600, 442225, 443556, 448900, 455625, 459684, 462400, 465124, 467856, 474552, 476100, 480249, 492804, 501264, 504100, 508369, 509796, 512000, 518400, 525625, 544644, 558009, 562500, 571536, 577600, 585225, 589824, 592704, 592900, 599076, 608400, 613089, 614656, 617796, 627264, 640000, 648025, 656100, 658503, 664225, 665856, 670761, 675684, 685584, 695556, 700569, 705600, 715716, 729000, 729316, 739600, 746496, 762129, 767376, 774400, 777924, 788544, 799236, 804357, 810000, 813604, 819025, 826281, 842724, 853776, 876096, 883600, 884736, 887364, 893025, 898704, 910116, 921600, 944784, 956484, 962361, 968256, 970225, 972196, 980100, 984064, 990025, 998001, 1004004, 1008016, 1010025, 1014049, 1016064, 1020100, 1022121, 1024144, 1034289.

Some larger examples :