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aban n.  and  powerful n.

First numbers that belong to both sets:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000000, 1000188, 2000000, 2000376, 3000564, 4000000, 4000752, 5000000, 5000211, 8000000, 9000000, 10000000, 15000129, 15000633, 16000000, 18000000, 19000881, 20000000, 20000844, 24000201, 25000000, 27000000, 31000500, 32000000, 34000561, 36000000, 39000025, 40000000, 45000000, 45000387, 49000000, 50000000, 54000000, 60000516, 64000000, 72000000, 72000603, 77000625, 80000000, 81000000, 81000992, 89000356, 90000000, 96000804, 98000000, 100000000, 108000000, 121000000, 125000000, 126000625, 128000000, 135000000, 144000000, 156000100, 157000900, 160000000, 162000000, 169000000, 178000712, 180000000, 195000125, 195000988, 196000000, 200000000, 216000000, 225000000, 242000000, 243000000, 245000000, 250000000, 251000649, 253000836, 256000000, 270000000, 282000500, 288000000, 289000000, 305000667, 307000368, 312000200, 313000200, 320000000, 324000000, 335000419, 338000000, 343000000, 351000225, 360000000, 361000000, 392000000, 392000401, 399000625, 400000000, 401000625, 405000000, 414000409, 427000896, 432000000, 441000000, 450000000, 484000000, 486000000, 490000000, 499000500, 500000000, 501000500, 512000000, 529000000, 540000000, 576000000, 578000000, 585000736, 605000000, 614000736, 624000400, 625000000, 626000400, 640000000, 648000000, 659000241, 675000000, 676000000, 686000000, 704000089, 720000000, 722000000, 729000000, 780000500, 784000000, 800000000, 810000000, 828000625, 841000000, 844000587, 845000000, 864000000, 882000000, 896000917, 900000000, 939000600, 946000125, 961000000, 968000000, 972000000, 975000625, 976000081, 977000049, 978000529, 980000000, 1000000000, 1024000000, 1044000721, 1053000675, 1058000000, 1061000329, 1080000000, 1089000000, 1097000641, 1123000832, 1125000000, 1152000000, 1156000000, 1210000000, 1215000000, 1225000000, 1248000800, 1249000200, 1250000000, 1251000200, 1252000800, 1268000881, 1280000000, 1283000761, 1296000000, 1323000000, 1331000000, 1350000000, 1352000000, 1369000000, 1372000000, 1375000561, 1404000900, 1407000100, 1440000000, 1444000000, 1445000000, 1451000464, 1458000000, 1521000000, 1531000384, 1533000500, 1568000000, 1583000523, 1598000625, 1600000000, 1602000625, 1620000000, 1681000000, 1682000000, 1690000000, 1715000000, 1728000000, 1764000000, 1775000304, 1783000328, 1800000000, 1805000000, 1849000000, 1900000921, 1922000000, 1928000281, 1936000000, 1944000000, 1957000644, 1960000000, 1977000723, 1988000569, 1998000500, 1998000601, 1999000125, 2000000000, 2001000125, 2002000500, 2025000000, 2048000000, 2116000000, 2131000328.

Some larger examples :