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arithmetic n.  and  perfect power

First numbers that belong to both sets:

1, 27, 49, 125, 169, 343, 361, 961, 1331, 1369, 1849, 2187, 2197, 2744, 3125, 3375, 3721, 4489, 4913, 5329, 6241, 6859, 7776, 8281, 9261, 9409, 10609, 11881, 12167, 13824, 14641, 16129, 16807, 17689, 19321, 22801, 24389, 24649, 26569, 29791, 32761, 35937, 37249, 39601, 42875, 44521, 47089, 49729, 50653, 52441, 58081, 59319, 61009, 67081, 68921, 73441, 74088, 76729, 78125, 79507, 80089, 87616, 90601, 94249, 97336, 97969, 103823, 109561, 113569, 121801, 132651, 134689, 139129, 140608, 143641, 148877, 157609, 161051, 162409, 166375, 167281, 175616, 177241, 182329, 185193, 187489, 192721, 205379, 208849, 214369, 219961, 226981, 231361, 237169, 238328, 249001, 261121, 273529, 274625, 287296, 287496, 292681, 299209, 300763, 305809, 312481, 326041, 328509, 332929, 343000, 346921, 357911, 361201, 368449, 371293, 375769, 383161, 389017, 398161, 399424, 413449, 436921, 452929, 456533, 461041, 477481, 493039, 494209, 502681, 512000, 519841, 528529, 537289, 537824, 546121, 564001, 571787, 573049, 582169, 591361, 614125, 619369, 628849, 657721, 658503, 667489, 677329, 687241, 704969, 717409, 727609, 737881, 753571, 758641, 759375, 760384, 769129, 779689, 790321, 804357, 822649, 823543, 830584, 844561, 857375, 877969, 900601, 912673, 923521, 935089, 946729, 982081, 994009, 998001, 1018081, 1030301, 1042441, 1054729, 1067089, 1079521, 1092727, 1104601, 1117249, 1129969, 1142761, 1157625, 1181569, 1194649, 1207801, 1225043, 1247689, 1261129, 1274641, 1295029, 1301881, 1315609, 1329409, 1343281, 1367631, 1371241, 1419857, 1442401, 1442897, 1459264, 1471369, 1481544, 1515361, 1520875, 1530169, 1560001, 1590121, 1601613, 1605289, 1620529, 1635841, 1666681, 1682209, 1685159, 1697809, 1700416, 1728000, 1745041, 1760929, 1776889, 1792921, 1825201, 1860867, 1907161, 1923769, 1940449, 1957201, 2007889, 2024929, 2042041, 2048383, 2093809, 2111209, 2128681, 2146689, 2163841, 2181529, 2199289, 2217121, 2248091, 2253001, 2343961, 2352637, 2380849, 2383936, 2399401, 2436721, 2455489, 2474329, 2476099, 2493241, 2531281, 2550409, 2569609, 2571353, 2588881, 2627641, 2628072, 2647129, 2685619, 2725801, 2745649, 2765569, 2785561, 2803221, 2825761, 2845969, 2863288, 2866249, 2886601, 2924207, 2968729, 2989441, 3031081, 3048625, 3052009.

Some larger examples :