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happy n.  and  hexagonal n.

First numbers that belong to both sets:

1, 28, 91, 190, 496, 946, 1128, 3321, 8128, 9730, 11476, 12090, 13366, 15753, 16471, 17205, 17578, 21115, 22791, 27261, 30135, 31626, 33670, 35245, 41328, 43660, 44253, 56280, 56953, 58311, 69751, 75855, 79003, 84666, 93961, 100128, 101025, 102831, 105570, 107416, 110215, 122760, 123753, 133903, 142311, 146611, 147696, 148785, 156520, 162165, 163306, 172578, 174936, 177310, 183315, 194376, 204480, 213531, 226801, 240471, 244650, 246051, 257403, 260281, 261726, 270480, 299151, 311655, 313236, 321201, 327645, 340725, 345696, 357435, 365940, 381501, 388521, 395605, 400960, 402753, 404550, 409965, 430128, 458403, 481671, 489555, 499500, 507528, 515620, 540280, 542361, 622170, 646953, 653796, 672220, 686206, 700336, 709836, 714610, 724206, 733866, 748476, 775635, 803278, 818560, 833986, 844350, 867903, 873181, 918690, 929566, 946000, 951510, 954271, 1001820, 1013176, 1030330, 1033203, 1038961, 1053426, 1103355, 1115271, 1142316, 1160526, 1185030, 1188111, 1194285, 1212903, 1234806, 1241100, 1253736, 1279200, 1301691, 1308153, 1327635, 1360425, 1390278, 1417086, 1447551, 1533876, 1547920, 1579753, 1583310, 1615503, 1651653, 1669878, 1706628, 1781328, 1796460, 1800253, 1869211, 1959210, 1979055, 2011015, 2027091, 2031120, 2039190, 2051325, 2055378, 2096128, 2104326, 2158003, 2162160, 2166321, 2183005, 2187186, 2293011, 2318781, 2327403, 2362051, 2392578, 2410110, 2471976, 2489796, 2530125, 2534626, 2539131, 2588950, 2607186, 2717946, 2750685, 2764776, 2797795, 2812006, 2831010, 2869210, 2893215, 2912491, 2956096, 2960961, 2965830, 2970703, 3014740, 3024570, 3029491, 3064050, 3078921, 3123750, 3138765, 3184026, 3194128, 3199185, 3214380, 3280641, 3316600, 3342405, 3420420, 3488761, 3499335, 3531153, 3579150, 3670695, 3725085, 3779875, 3812941, 3818466, 3896236, 3913003, 3946645, 3957891, 4014361, 4025703, 4180386, 4197753, 4203550, 4250070, 4403028, 4474536, 4510506, 4516515, 4546620, 4576825, 4631446, 4717056, 4735503, 4753986, 4803450, 4846941, 4865640, 4959675, 4972281, 5010195, 5054610, 5060971, 5092836, 5240703, 5299140, 5305653, 5351356, 5357901, 5377560, 5423571, 5443350, 5456556, 5469778, 5489641, 5542785, 5562780, 5629690, 5643120, 5649841, 5656566, 5724036, 5764710, 5867025, 5880735, 5894461, 5942628, 5977153, 6018715, 6032601, 6039550, 6067386, 6123250, 6158295, 6327903, 6550390, 6564876, 6579378, 6798828.

Some larger examples :