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binomial coefficient  and  happy n.

First numbers that belong to both sets:

10, 28, 70, 91, 190, 496, 680, 820, 946, 1128, 1275, 1330, 1771, 2080, 2211, 2300, 2485, 3321, 3654, 4278, 4960, 6435, 8128, 8256, 8778, 9591, 9730, 11476, 11480, 12090, 12341, 12880, 13203, 13366, 13530, 15180, 15753, 16471, 17205, 17550, 17578, 20475, 20825, 20910, 21115, 21321, 22791, 24753, 24804, 25651, 27261, 27405, 29890, 30135, 31626, 31824, 33670, 35245, 36046, 41328, 43660, 43956, 44253, 46360, 47586, 48205, 50721, 53130, 53301, 53956, 54264, 54615, 55278, 56280, 56953, 58311, 61425, 62128, 66430, 67525, 69378, 69751, 70125, 75582, 75855, 76245, 77815, 79003, 80200, 81810, 82621, 84666, 87571, 90100, 90951, 93961, 95284, 99681, 100128, 101025, 102831, 103285, 105570, 107416, 109736, 110215, 117370, 119316, 122760, 123256, 123410, 123753, 126253, 127260, 129286, 130305, 133903, 137550, 139656, 142311, 142506, 146611, 147696, 148785, 150426, 156520, 157080, 158203, 162165, 163306, 165025, 167960, 169653, 171700, 172578, 174936, 176851, 177100, 177310, 179101, 181503, 182104, 183315, 194376, 197506, 203841, 204156, 204480, 210276, 213531, 218130, 226801, 228826, 230181, 234270, 235641, 238395, 240471, 244650, 245350, 246051, 246753, 249571, 257403, 260281, 261726, 270480, 273819, 280126, 287661, 287980, 295240, 296835, 299151, 303031, 309291, 310124, 311655, 313236, 314028, 315615, 317750, 318801, 321201, 327645, 336610, 340725, 345696, 356590, 357435, 358281, 363378, 365940, 373680, 381501, 384126, 388521, 392941, 394716, 395605, 400960, 400995, 402753, 404550, 405450, 407253, 409060, 409965, 416328, 425503, 427350, 430128, 434778, 457310, 458403, 475020, 478731, 480700, 481671, 489555, 499500, 506521, 507528, 514605, 515620, 518671, 520710, 524800, 529396, 530965, 540280, 542361, 553878, 568711, 570846, 585276, 596904, 621055, 622170, 646646, 646953, 653796, 672220, 673380, 686206, 700336, 701520, 705432, 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, 906192, 911925, 918690, 925480, 929566, 946000.

Some larger examples :