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happy n.  and  octagonal n.

First numbers that belong to both sets:

1, 133, 176, 280, 736, 833, 1281, 1825, 1976, 2133, 3605, 6533, 11041, 11408, 20008, 25208, 26885, 27456, 28033, 34133, 35425, 36080, 52801, 53600, 54405, 55216, 56033, 69921, 70840, 78408, 80360, 83333, 85345, 88408, 99008, 110208, 111361, 120801, 123221, 131880, 165205, 170885, 200725, 203841, 205408, 208560, 211736, 226325, 253461, 262256, 282133, 300833, 314280, 316225, 320133, 328021, 330008, 332001, 336005, 344085, 352261, 375240, 385925, 446216, 448533, 469656, 476805, 553840, 587861, 653333, 687365, 698901, 704705, 752001, 773176, 776225, 791560, 819541, 835296, 841640, 848008, 883461, 889985, 893256, 906400, 956545, 959936, 987280, 1004565, 1015008, 1043120, 1050208, 1060885, 1064456, 1082401, 1111425, 1118741, 1140833, 1159408, 1185665, 1200801, 1262305, 1313408, 1460216, 1481221, 1515141, 1519408, 1592865, 1597240, 1601621, 1650208, 1672533, 1704040, 1749560, 1754145, 1786408, 1800325, 1814296, 1823640, 1851816, 1861256, 1870720, 1875461, 1884961, 1889720, 1923201, 1937640, 1952133, 1986160, 2000833, 2010645, 2015560, 2100033, 2105056, 2165800, 2258536, 2274181, 2493408, 2515336, 2570576, 2576133, 2615200, 2620805, 2626416, 2717008, 2962120, 2980033, 2998000, 3040133, 3119160, 3131408, 3155976, 3242720, 3261461, 3311701, 3330640, 3375041, 3400545, 3490565, 3516501, 3542533, 3549056, 3562120, 3575208, 3621205, 3667496, 3674133, 3687425, 3734136, 3747536, 3774408, 3876033, 3896520, 3923920, 3999765, 4020576, 4048408, 4125441, 4189008, 4253061, 4346440, 4375376, 4440833, 4455445, 4499425, 4617761, 4700008, 4745176, 4775408, 4843781, 4859041, 4897296, 4997461, 5020720, 5059605, 5106465, 5145680, 5312021, 5328001, 5336000, 5344005, 5352016, 5376085, 5400208, 5416320, 5440533, 5456705, 5472901, 5489121, 5513496, 5521633, 5603333, 5611536, 5669125, 5718721, 5793520, 6046360, 6261185, 6304600, 6339440, 6435745, 6497408, 6532776, 6550496, 6639456, 6684161, 6693120, 6837280, 6855408, 6873560, 7139461, 7223008, 7288325, 7335160, 7363333, 7457633, 7476565, 7524000, 7763425, 7792408, 7850536, 7889408, 7899141, 7947896, 7957665, 8045856, 8055685, 8134533, 8213765, 8243576, 8263480, 8313345, 8383408, 8423576, 8504200, 8564920, 8646216, 8666600, 8697221, 8789408, 8799681, 8809960, 8840833, 8871760, 8892408, 8913080, 8923425, 8954496, 8975240, 8996008, 9173505, 9384545, 9405781, 9427041, 9490965, 9662485, 9727201.