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triangular n.  and  untouchable n.

First numbers that belong to both sets:

120, 210, 276, 406, 1128, 1326, 1596, 1830, 2628, 2850, 3486, 3828, 5778, 5886, 6216, 7140, 7260, 7750, 8778, 9180, 10296, 10440, 11628, 12880, 13530, 14706, 15400, 16836, 17766, 18336, 19110, 20100, 20910, 21528, 23436, 25200, 25878, 27966, 28680, 28920, 29646, 30628, 30876, 32640, 33670, 34716, 34980, 35778, 36046, 37128, 38226, 39060, 41328, 41616, 44850, 47586, 48516, 53628, 53956, 57970, 60726, 61776, 64620, 64980, 66066, 67528, 69006, 70500, 72390, 73536, 81406, 83028, 83436, 84666, 86736, 87990, 88410, 90100, 91378, 95266, 98790, 102378, 103740, 104196, 107880, 111156, 111628, 113050, 115440, 119316, 122760, 123256, 126756, 127260, 131328, 134940, 135460, 137550, 139656, 143916, 146070, 150426, 154290, 159330, 163878, 165600, 170820, 174936, 177906, 179700, 180300, 182106, 185136, 190036, 194376, 196878, 200028, 201930, 202566, 207046, 209628, 210276, 214840, 215496, 217470, 223446, 225456, 226128, 228150, 228826, 231540, 233586, 237016, 239778, 241860, 242556, 244650, 250278, 253116, 256686, 258840, 259560, 261726, 264628, 265356, 276396, 277140, 280126, 282376, 283128, 286146, 294528, 297606, 298378, 300700, 301476, 303810, 304590, 306936, 307720, 310866, 313236, 322806, 330078, 332520, 333336, 335790, 336610, 339900, 343206, 346528, 355746, 359128, 359976, 362526, 363378, 365940, 372816, 383250, 386760, 391170, 393828, 394716, 397386, 400960, 404550, 405450, 408156, 411778, 412686, 415416, 423660, 437580, 438516, 442270, 446040, 448878, 449826, 453628, 457446, 461280, 465130, 468996, 484620, 487578, 495510, 496506, 499500, 500500, 503506, 507528, 508536, 516636, 523776, 531996, 533028, 536130, 540280, 544446, 545490, 548628, 557040, 558096, 561270, 570846, 574056, 579426, 583740, 588070, 592416, 595686, 600060, 605550, 608856, 609960, 618828, 623286, 626640, 627760, 636756, 640146, 645816, 650370, 653796, 662976, 673380, 676866, 690900, 692076, 701520, 719400, 729028, 730236, 733866, 735078, 748476, 749700, 763230, 768180, 779376, 783126, 789396, 793170, 808356, 809628, 819840, 824970, 835278, 839160, 844350, 856086, 860016, 865270, 866586, 871860, 882456, 891780, 898470, 902496, 903840, 913276, 914628, 920046, 924120, 935028, 952890, 957036, 958420, 963966, 986310, 990528.