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balanced n.  and  octagonal n.

First numbers that belong to both sets:

21, 225, 936, 2296, 3605, 8321, 10680, 11041, 12545, 13333, 13736, 14145, 14981, 15841, 24120, 26320, 32240, 32865, 34776, 36080, 36741, 38760, 39445, 40833, 42960, 45880, 56856, 57685, 60208, 61920, 62785, 63656, 128133, 144760, 161008, 162401, 169456, 170885, 179585, 184016, 190008, 193040, 205408, 208560, 210145, 211736, 214936, 219781, 223041, 226325, 227976, 236321, 238008, 239701, 244816, 253461, 255208, 258720, 264033, 306560, 326040, 350208, 366800, 371008, 379496, 396760, 409960, 418880, 446216, 455520, 548696, 556421, 564200, 587861, 617440, 620165, 625633, 644960, 650536, 653333, 687365, 696008, 701800, 713456, 734085, 737056, 743016, 752001, 761040, 773176, 825825, 851201, 864033, 880208, 909701, 922965, 932976, 943041, 946408, 963333, 990725, 1004565, 1008040, 1018501, 2227408, 2279408, 2316165, 2321440, 2374520, 2471576, 2537360, 2548408, 2576133, 2587265, 2598421, 2609601, 2637656, 2711301, 2728440, 2734165, 2809136, 2844080, 2873365, 2914616, 2938320, 2944261, 2980033, 3022040, 3040133, 3046176, 3058280, 3076481, 3094736, 3119160, 3242720, 3255208, 3324321, 3336965, 3362325, 3368680, 3387781, 3394160, 3438981, 3451841, 3471176, 3516501, 3523000, 3536016, 3568661, 3575208, 3581761, 3601456, 3621205, 3647621, 3747536, 3781141, 3787880, 3808133, 3835221, 3917061, 3937656, 3951416, 3992840, 4020576, 4048408, 4076336, 4139525, 4146576, 4153633, 4245920, 4382625, 4536240, 4670016, 4782981, 4874325, 4889633, 4928008, 4935701, 5020720, 5106465, 5129976, 5185045, 5216645, 5296065, 5304040, 5368056, 5392161, 5416320, 5529776, 5677376, 5785185, 5961480, 6029336, 6295905, 6330721, 6348165, 6479760, 6612705, 6729016, 6765008, 6783040, 6801096, 6828225, 6882645, 7001296, 7019640, 7157985, 7269633, 7400981, 7438725, 7638456, 7676800, 7840833, 8006600, 8055685, 8075361, 8154305, 8164200, 8243576, 8253525, 8313345, 8333333, 8423576, 8514305, 8697221, 8748376, 8779141, 8809960, 8820245, 8830536, 8861445, 8871760, 8902741, 8923425, 9027205, 9100208, 9163016, 9194501, 9268176, 9299841, 9342145, 9363333, 9405781, 9427041, 9501640, 9533701, 9662485, 9694816, 9759640, 9824680, 9835541, 9879045, 9922645, 10010133, 10054021, 10087000, 10120033, 10164161, 10208385, 10252705, 10297121, 10319365, 10352776, 10386241, 10397408, 10408581, 10453333, 10498181, 10712520, 10735208, 10837601, 10894696, 10951941, 10997845, 11136133, 11194008, 11228805, 11356856, 11415301, 11438721.

Some larger examples :