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arithmetic n.  and  powerful n.

First numbers that belong to both sets:

1, 27, 49, 125, 169, 343, 361, 432, 500, 864, 961, 1125, 1323, 1331, 1369, 1849, 2187, 2197, 2744, 3125, 3375, 3721, 4489, 4563, 4913, 5324, 5329, 5488, 6125, 6241, 6859, 7776, 8281, 9261, 9409, 9747, 10609, 10976, 11881, 11979, 12167, 12500, 13500, 13824, 14641, 15125, 16129, 16807, 16875, 17689, 19321, 19652, 21125, 21168, 21600, 22801, 24389, 24500, 24649, 24696, 25947, 26569, 27783, 28125, 29791, 30375, 32761, 33275, 34992, 35152, 35937, 36125, 36963, 37249, 39601, 42336, 42592, 42875, 44217, 44521, 45125, 47089, 48668, 49729, 49923, 50653, 52441, 54000, 55125, 57967, 58081, 59319, 61009, 64827, 65219, 66125, 67081, 68600, 68921, 69984, 73008, 73441, 74088, 76729, 78125, 78608, 79507, 80089, 84375, 84500, 87616, 90601, 94249, 97336, 97556, 97969, 98784, 100467, 103823, 104544, 105125, 107163, 107653, 108000, 109503, 109561, 109744, 113569, 120125, 121203, 121801, 122825, 123823, 132651, 134456, 134689, 139129, 140608, 143641, 143748, 143883, 146016, 148176, 148877, 153125, 155952, 157609, 161051, 162409, 165375, 166375, 167281, 168507, 171125, 171500, 175616, 176868, 177241, 177957, 180500, 182329, 185193, 187489, 190125, 192721, 194672, 200000, 205379, 208849, 210125, 214369, 214375, 219488, 219501, 219961, 222264, 223587, 224939, 226981, 231125, 231361, 237169, 238328, 240737, 249001, 249696, 254043, 260876, 261121, 273375, 273529, 274400, 274625, 275684, 276125, 286443, 287296, 287496, 292681, 296352, 299209, 300763, 304175, 305809, 311904, 312481, 312500, 320787, 323433, 326041, 328509, 329623, 332024, 332929, 336091, 337500, 343000, 345600, 346921, 351125, 357911, 361201, 364500, 368449, 369603, 371293, 375769, 378125, 381024, 383161, 383328, 384659, 385875, 389017, 389344, 390224, 395307, 397953, 398161, 399424, 406125, 408375, 413449, 415152, 415292, 435125, 435483, 436921, 444528, 452929, 453789, 456533, 457056, 461041, 463736, 465125, 469567, 477481, 477603, 480491, 480500, 491300, 493039, 494209, 502681, 512000, 519841, 521667, 528125, 528529, 530604, 537289, 537824, 540000, 546121, 555579, 561125, 564001, 570375, 571787.

Some larger examples :