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balanced n.  and  triangular n.

First numbers that belong to both sets:

10, 15, 21, 78, 120, 153, 210, 528, 595, 630, 666, 820, 990, 1830, 2278, 2701, 2926, 7503, 8001, 8385, 9045, 9453, 9870, 10585, 11476, 12403, 12720, 13366, 13530, 13861, 14365, 14535, 15931, 17955, 18721, 18915, 23005, 23436, 28920, 29890, 30876, 31878, 32385, 34191, 34716, 35245, 36046, 37675, 37950, 39340, 39621, 40470, 40755, 41905, 42195, 42778, 43365, 44551, 45150, 45451, 46665, 48516, 49455, 50086, 50403, 51681, 52003, 52326, 54946, 55945, 56616, 57630, 57970, 59685, 62128, 128271, 131841, 136503, 138601, 140715, 145530, 146070, 149331, 149878, 150426, 154290, 158766, 159330, 159895, 162735, 167331, 173166, 174936, 175528, 176715, 185136, 185745, 190036, 194376, 200028, 201295, 202566, 203203, 207690, 210276, 214840, 216153, 219453, 220780, 222778, 224785, 226128, 228826, 230860, 233586, 236328, 241860, 242556, 248865, 250278, 250986, 253116, 258121, 294528, 299151, 310866, 315615, 318801, 330078, 332520, 334971, 335790, 343206, 349030, 352380, 379756, 396495, 405450, 415416, 419070, 427350, 434778, 440391, 443211, 448878, 449826, 456490, 464166, 465130, 468996, 483636, 488566, 492528, 493521, 495510, 500500, 503506, 507528, 527878, 535095, 538203, 540280, 553878, 564453, 567645, 568711, 578350, 579426, 588070, 595686, 598965, 606651, 615495, 616605, 625521, 626640, 634501, 657231, 687378, 689725, 694431, 696790, 699153, 709836, 719400, 727821, 730236, 731445, 733866, 736291, 741153, 743590, 757065, 764466, 770661, 773146, 779376, 785631, 791911, 799480, 809628, 812175, 814726, 816003, 828828, 833986, 837865, 857395, 862641, 866586, 869221, 889111, 891780, 893116, 897130, 924120, 925480, 926841, 930930, 936396, 943251, 947376, 948753, 951510, 952890, 968136, 970921, 984906, 986310, 1011753, 1016025, 1020306, 1024596, 1040403, 2137278, 2141415, 2160081, 2170486, 2176741, 2185095, 2208151, 2216565, 2220778, 2222886, 2246140, 2267385, 2271646, 2273778, 2275911, 2280180, 2288730, 2293011, 2320935, 2327403, 2333880, 2340366, 2342530, 2353365, 2375110, 2381653, 2388205, 2399145, 2405721, 2421100, 2429910, 2432115, 2443155, 2447578, 2454220, 2460871, 2463090, 2483106, 2494261, 2505441, 2516646, 2525628, 2534626, 2541385, 2543640, 2545896.

Some larger examples :