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balanced n.  and  powerful n.

First numbers that belong to both sets:

9, 49, 108, 169, 216, 225, 500, 625, 675, 676, 841, 961, 968, 2744, 2916, 3087, 3125, 3249, 3267, 3528, 3721, 3888, 3969, 4000, 7225, 8649, 10609, 11881, 11907, 11979, 12100, 12168, 12321, 12769, 13924, 14641, 14792, 15625, 16129, 16875, 17689, 22707, 27783, 29241, 32400, 36125, 38025, 39601, 40500, 41209, 41503, 42849, 44944, 45369, 45387, 45796, 47524, 52900, 54000, 54872, 55112, 56644, 58081, 60500, 60516, 61731, 62001, 64009, 64800, 65025, 128547, 133100, 136107, 137200, 151321, 154449, 154568, 155236, 156025, 158184, 160083, 161604, 163216, 163592, 167281, 168507, 176868, 177241, 177608, 180000, 181447, 182408, 183184, 185761, 187272, 187489, 189225, 191664, 192721, 194672, 195075, 195112, 198916, 199712, 207576, 209764, 212552, 217156, 218700, 219024, 219700, 220448, 223112, 225000, 228484, 228488, 231361, 232324, 236196, 236672, 239432, 240737, 244036, 245000, 247009, 247808, 250000, 252004, 253472, 254043, 255025, 256328, 259081, 260876, 261121, 264600, 276768, 288000, 288800, 289000, 295936, 300304, 302500, 304704, 311904, 314721, 315844, 316368, 326432, 336400, 338688, 345744, 364816, 366368, 372100, 383688, 384400, 386884, 396508, 401956, 412164, 414736, 419528, 432000, 438012, 441800, 446224, 453152, 456300, 463736, 480200, 490000, 496008, 531441, 533871, 544500, 561001, 571787, 583443, 596183, 609961, 613089, 613832, 615627, 622521, 630436, 632025, 636804, 644204, 649636, 653400, 660969, 665500, 667489, 674041, 675684, 677329, 684500, 686792, 687241, 696200, 701784, 702244, 707281, 711828, 712336, 714025, 715208, 717409, 724201, 727609, 731025, 740772, 741125, 744012, 744769, 748225, 751689, 751896, 767376, 768800, 771147, 773768, 777600, 781456, 784000, 810448, 817452, 820125, 829921, 830584, 837225, 840500, 850084, 855468, 857500, 861125, 877969, 882000, 885481, 894348, 898704, 902500, 904203, 908552, 910116, 914112, 924500, 927472, 933156, 940896, 946729, 953312, 960400, 963272, 972000, 980100, 986049, 996004, 998784, 1012036, 1016172, 1018081, 1028196, 1028376, 1034289, 1042441, 1042568, 1046529, 2074464, 2105352, 2112500, 2131947, 2145125, 2171528, 2196608, 2208492, 2211169, 2221832, 2222316.

Some larger examples :