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centered nonagonal n.  and  balanced n.

First numbers that belong to both sets:

10, 595, 820, 2278, 2701, 2926, 10585, 11476, 12403, 13366, 13861, 14365, 15931, 18721, 23005, 29890, 35245, 36046, 37675, 39340, 41905, 42778, 44551, 45451, 50086, 52003, 54946, 55945, 57970, 62128, 138601, 149878, 159895, 175528, 190036, 201295, 203203, 214840, 220780, 222778, 224785, 228826, 230860, 258121, 349030, 379756, 456490, 465130, 488566, 500500, 503506, 527878, 540280, 568711, 578350, 588070, 634501, 689725, 696790, 736291, 743590, 773146, 791911, 799480, 814726, 833986, 837865, 857395, 869221, 889111, 893116, 897130, 925480, 970921, 2170486, 2176741, 2208151, 2220778, 2246140, 2271646, 2342530, 2375110, 2381653, 2388205, 2421100, 2447578, 2454220, 2460871, 2494261, 2534626, 2541385, 2561716, 2582128, 2713285, 2727280, 2762425, 2797795, 2812006, 2905255, 2926990, 2941525, 2978020, 2992681, 3036880, 3096316, 3163870, 3224530, 3262735, 3270403, 3347578, 3355345, 3417805, 3425653, 3457135, 3504628, 3520531, 3568456, 3673405, 3689686, 3722356, 3738745, 3763396, 3788128, 3862810, 3871153, 3879505, 3946645, 3955078, 4022866, 4031380, 5128003, 5371003, 5449951, 5539456, 5619628, 5720653, 5751136, 6175855, 6186403, 6356395, 6420736, 6714280, 6769360, 6780403, 6835753, 7930153, 8134561, 8489260, 8501626, 8600878, 8638246, 8738290, 8952796, 9003646, 9016381, 9067411, 9105778, 9118585, 9157060, 9247150, 9298828, 9324721, 9337681, 9389611, 9480835, 9493903, 9559378, 9651421, 9730666, 9783676, 9850141, 9930196, 9943570, 10010575, 10050886, 10064341, 10118251, 10131751, 10199386, 10253656, 10294453, 10389961, 10431028, 10499653, 10860130, 10902115, 10986328, 11056753, 11070865, 11113255, 11169901, 11226691, 11269378, 11412253, 11527201, 11570455, 11686195, 11773378, 11860885, 11919403, 11948716, 11978065, 12007450, 12140128, 12154915, 12169711, 12288403, 12333061, 12437578, 12542536, 12632851, 12647935, 12768931, 13027960, 13089286, 13135375, 13212370, 13227796, 13289590, 13320541, 13382551, 13429153, 13460266, 13491415, 13522600, 13647700, 13726180, 13804885, 13899628, 13915450, 13931281, 13947121, 13978828, 14074165, 14121955, 14169826, 14185801, 14217778, 14265811, 14297878, 14734306, 14783203, 14963185, 14996026, 15078286, 15111253, 15243481, 15260050, 16253551, 16304905, 16390675, 16425046, 16597441, 33919966, 33969403, 33994135, 34068385, 34167511, 34815340, 35393491, 35747740, 35976403, 36001855, 36359128, 36795331, 36821071, 37001503, 37130653, 37234135, 37260028, 37337761, 38041003, 38381941, 38487151, 38618866, 38698003, 38750806, 38777221, 38882971, 39094903.

Some larger examples :