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happy n.  and  triangular n.

First numbers that belong to both sets:

1, 10, 28, 91, 190, 496, 820, 946, 1128, 1275, 2080, 2211, 2485, 3321, 4278, 8128, 8256, 8778, 9591, 9730, 11476, 12090, 12880, 13203, 13366, 13530, 15753, 16471, 17205, 17578, 20910, 21115, 21321, 22791, 24753, 25651, 27261, 29890, 30135, 31626, 33670, 35245, 36046, 41328, 43660, 43956, 44253, 46360, 47586, 48205, 50721, 53301, 53956, 54615, 55278, 56280, 56953, 58311, 61425, 62128, 66430, 69378, 69751, 70125, 75855, 76245, 77815, 79003, 80200, 81810, 82621, 84666, 87571, 90100, 90951, 93961, 99681, 100128, 101025, 102831, 103285, 105570, 107416, 110215, 117370, 119316, 122760, 123256, 123753, 126253, 127260, 129286, 130305, 133903, 137550, 139656, 142311, 146611, 147696, 148785, 150426, 156520, 157080, 158203, 162165, 163306, 165025, 169653, 172578, 174936, 177310, 179101, 181503, 183315, 194376, 197506, 203841, 204480, 210276, 213531, 218130, 226801, 228826, 230181, 234270, 235641, 238395, 240471, 244650, 245350, 246051, 246753, 249571, 257403, 260281, 261726, 270480, 280126, 287661, 296835, 299151, 303031, 309291, 311655, 313236, 314028, 315615, 318801, 321201, 327645, 336610, 340725, 345696, 356590, 357435, 358281, 363378, 365940, 373680, 381501, 384126, 388521, 392941, 394716, 395605, 400960, 402753, 404550, 405450, 407253, 409060, 409965, 416328, 425503, 427350, 430128, 434778, 458403, 478731, 481671, 489555, 499500, 506521, 507528, 514605, 515620, 518671, 520710, 524800, 530965, 540280, 542361, 553878, 568711, 570846, 621055, 622170, 646953, 653796, 672220, 673380, 686206, 700336, 701520, 708645, 709836, 714610, 724206, 733866, 748476, 766941, 775635, 776881, 786885, 789396, 799480, 802011, 803278, 818560, 819840, 822403, 832695, 833986, 844350, 867903, 869221, 873181, 877150, 893116, 895791, 911925, 918690, 925480, 929566, 946000, 951510, 954271, 958420, 969528, 975106, 983503, 1001820, 1011753, 1013176, 1026028, 1028895, 1030330, 1033203, 1034641, 1038961, 1053426, 1054878, 1060696, 1075311, 1087075, 1098903, 1103355, 1115271, 1131760, 1142316, 1149886, 1160526, 1171215, 1185030, 1186570, 1188111, 1194285, 1211346, 1212903, 1223830, 1234806, 1241100, 1253736, 1255320, 1258491, 1279200, 1280800, 1284003, 1293636, 1296855.

Some larger examples :