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alternating n.  and  perfect power

First numbers that belong to both sets:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 81, 121, 125, 169, 216, 256, 343, 361, 529, 676, 729, 961, 1296, 2187, 4761, 5476, 5832, 6561, 7056, 9216, 12167, 12321, 12769, 14161, 16129, 18769, 32761, 34969, 41616, 56169, 69696, 72361, 74529, 76729, 78125, 78961, 87616, 96721, 147456, 163216, 181476, 212521, 214369, 290521, 329476, 341056, 367236, 436921, 452929, 458329, 474721, 527076, 541696, 583696, 614125, 658503, 670761, 690561, 763876, 781456, 929296, 1030301, 1038361, 1054729, 1092727, 1234321, 1238769, 1252161, 1270129, 1292769, 1610361, 1630729, 1656369, 1852321, 1874161, 1896129, 2143296, 2509056, 2547216, 2903616, 3470769, 3876961, 3892729, 4145296, 4301476, 4343056, 4741632, 4769856, 5230369, 5414929, 5470921, 5612161, 5650129, 5812921, 5870929, 5890329, 7230721, 7414729, 7436529, 7458361, 7634169, 8503056, 8561476, 8741816, 8785296, 9078169, 9418761, 9492561, 9678321, 9690769, 10929636, 12503296, 14167696, 14561856, 16941456, 18181696, 18541636, 21058921, 21836929, 21874329, 25230529, 25270729, 25694761, 25816561, 27270901, 27678121, 27818127, 29278921, 29452329, 32307856, 32787076, 32901696, 34503876, 38365636, 38763076, 41254929, 41434969, 43072969, 43256929, 43414921, 45252529, 45414121, 47238129, 47430769, 47458321, 47832147, 47872561, 49434961, 50183056, 52765696, 54523456, 61418569, 63632529, 63696361, 69072721, 69272329, 70123876, 76143076, 76983076, 78747876, 78925456, 81234169, 81414529, 83010321, 87216921, 89056969, 89094721, 94167616, 94789696, 98327056, 98525476, 103652761, 105452361, 105616729, 105698961, 109432521, 109474369, 121418361, 123454321, 123498769, 123632161, 123810129, 125014761, 129072321, 129254161, 129436129, 143256961, 161010721, 161874729, 165456769, 167210761, 169494361, 181252369, 183250369, 218921616, 232989696, 250145856, 252301456, 303212569, 307616521, 321090561, 329894569, 343694521, 345476569, 347412321, 349278721, 361418121, 361494169, 361874529, 365230321, 367450561, 381078125, 381694369, 383258929, 383650569, 385690321, 387656721, 416323216, 436141456, 450543076, 450967696, 452583076, 474107076, 478909456, 492129856, 503418969, 521254561, 523814769, 525830761, 529874361, 541074121, 541632529, 541818729, 549292969, 567058969, 581436769, 583850569, 585010969, 587432169, 589858369, 654541056, 674129296, 690323076, 709210161, 709636321, 723018321, 725278761, 741854169, 743816529, 745890721, 763030129, 763472161, 769452121, 783272169, 814303296, 818303236, 830361856, 854743696, 856381696, 856967076, 894967056, 907636129, 927872521, 941078329, 941814721, 943656961, 949810761, 967894321, 987656329.