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ABA n.  and  perfect power

First numbers that belong to both sets:

8, 32, 64, 81, 128, 324, 512, 1024, 2048, 2187, 2500, 5184, 5832, 8192, 9604, 15625, 16384, 26244, 32768, 40000, 58564, 59049, 82944, 114244, 125000, 131072, 153664, 177147, 202500, 262144, 279936, 331776, 334084, 373248, 419904, 521284, 524288, 640000, 777924, 937024, 941192, 1048576, 1119364, 1265625, 1327104, 1562500, 1594323, 1827904, 1889568, 2097152, 2125764, 2359296, 2458624, 2829124, 3240000, 3694084, 3779136, 4194304, 4251528, 4743684, 5345344, 5764801, 6002500, 6718464, 7496644, 8000000, 8340544, 8388608, 9253764, 9529569, 10240000, 11303044, 12446784, 13675204, 14172488, 14992384, 16000000, 16402500, 17909824, 19518724, 21233664, 23059204, 23887872, 25000000, 27060804, 29246464, 31561924, 33554432, 34012224, 36602500, 38614472, 39337984, 42224004, 43046721, 45265984, 48469444, 48828125, 51840000, 55383364, 59105344, 60236288, 63011844, 67108864, 71402500, 75898944, 80604484, 81000000, 85525504, 90668484, 91125000, 96040000, 101646724, 107495424, 113592964, 119946304, 126562500, 133448704, 134217728, 140612164, 143496441, 148060224, 155800324, 163840000, 172186884, 180848704, 189833284, 193100552, 199148544, 208802500, 218803264, 229159044, 239878144, 241864704, 250968964, 262440000, 272097792, 274299844, 286557184, 299220804, 312299584, 312500000, 325802500, 339738624, 354117124, 368947264, 376367048, 384238404, 390971529, 400000000, 416241604, 432972864, 450203524, 467943424, 486202500, 504990784, 512000000, 524318404, 536870912, 544195584, 564632644, 585640000, 607228164, 609892416, 612220032, 629407744, 652189444, 675584064, 686128968, 688747536, 699602500, 724255744, 749554884, 775511104, 802135684, 829440000, 857435524, 886133824, 907039232, 915546564, 922640625, 945685504, 976562500, 1008189504, 1040578564, 1073741824, 1107691524, 1142440000, 1162261467, 1177999684, 1184287112, 1214383104, 1251602884, 1289671744, 1328602500, 1358954496, 1368408064, 1409101444, 1450695744, 1493204164, 1528823808, 1536640000, 1581016644, 1626347584, 1672646404, 1719926784, 1768202500, 1817487424, 1867795524, 1919140864, 1934917632, 1953125000, 1955143089, 1971537604, 2025000000, 2079542404, 2135179264, 2147483648, 2191925124, 2249794624, 2308802500, 2368963584, 2430292804, 2471326208, 2492805184, 2556515844, 2621440000, 2687592964, 2754990144, 2823647044, 2893579264, 2964802500, 3037332544, 3099363912, 3111185284, 3186376704, 3262922884, 3340840000, 3420144324, 3486784401, 3500852224, 3582980164, 3623878656, 3666544704, 3751562500, 3810716361, 3838050304, 3855122432, 3926024964, 4015503424, 4106502724, 4199040000, 4293132484, 4388797504, 4486052484, 4584914944, 4586471424, 4685402500, 4758586568, 4787532864, 4891323844, 4996793344, 5103959364, 5184000000, 5212840000, 5323453444, 5435817984, 5549952004, 5665873984, 5783602500, 5832000000, 5903156224, 6024553924, 6147814464, 6272956804, 6400000000, 6528963204, 6659865664, 6792726724, 6927565824, 6947055801.

Some larger examples :