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triangular n.  and  Ulam n.

First numbers that belong to both sets:

1, 3, 6, 28, 36, 253, 820, 861, 1035, 1081, 1431, 1770, 2628, 3916, 8128, 9870, 11781, 15400, 16653, 17391, 20301, 20910, 21736, 22791, 23871, 25425, 27495, 29403, 31878, 37950, 39903, 41041, 48516, 49141, 52650, 54285, 56616, 58996, 59340, 61075, 61776, 64980, 67528, 68635, 70876, 73920, 77028, 81810, 82621, 95266, 102831, 105570, 106953, 110215, 114960, 118828, 125250, 130305, 133386, 134940, 135981, 139128, 139656, 144453, 152076, 155961, 162165, 177310, 179101, 186966, 196251, 198135, 201930, 205761, 228150, 232221, 243253, 245350, 248160, 257403, 263175, 289180, 295296, 302253, 324415, 383250, 397386, 408156, 417241, 424581, 437580, 444153, 456490, 471906, 472878, 473851, 479710, 549676, 567645, 653796, 657231, 662976, 695610, 706266, 726615, 736291, 743590, 746031, 757065, 773146, 784378, 793170, 831405, 832695, 841753, 883785, 893116, 910575, 914628, 926841, 937765, 977901, 1001820, 1041846, 1049076, 1078246, 1100386, 1121253, 1133265, 1142316, 1180416, 1228528, 1272810, 1288815, 1352190, 1360425, 1370340, 1420455, 1452660, 1490401, 1493856, 1523385, 1558495, 1576200, 1590436, 1606528, 1638955, 1660753, 1708476, 1768140, 1875016, 1929630, 1971105, 1973091, 2007006, 2009010, 2033136, 2049300, 2051325, 2059435, 2094081, 2096128, 2116653, 2118711, 2122830, 2137278, 2149701, 2153850, 2191371, 2237670, 2271646, 2288730, 2293011, 2331720, 2333880, 2338203, 2403528, 2416701, 2454220, 2469753, 2471976, 2480878, 2512161, 2534626, 2541385, 2591226, 2646150, 2650753, 2697003, 2745996, 2760075, 2783620, 2835771, 2842920, 2859636, 2874003, 2883601, 2956096, 2982903, 2985346, 3076440, 3148795, 3151305, 3224530, 3326910, 3329490, 3337236, 3386503, 3478203, 3488761, 3496690, 3515226, 3552445, 3649051, 3692403, 3716901, 3785376, 3801903, 3843378, 3848925, 3871153, 3932610, 4000206, 4003035, 4022866, 4045590, 4068378, 4183278, 4186171, 4191960, 4250070, 4285128, 4302711, 4323270, 4364535, 4394130, 4462578, 4495501, 4522528, 4525536, 4598028, 4643628, 4695580, 4726275, 4812753, 4825171, 4853170, 4862521, 4881250, 4890628, 4909411, 4928230, 4991220, 5000703, 5038725, 5086455, 5144028, 5172936, 5182590, 5198700, 5201925, 5286126, 5344815, 5430160, 5456556, 5549446, 5562780, 5670028, 5740966, 5744355, 5856753, 5921961, 5946076, 5997916, 6049981, 6070870, 6091795, 6119751, 6123250, 6193440.

Some larger examples :