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happy n.  and  pentagonal n.

First numbers that belong to both sets:

1, 70, 176, 376, 1247, 1335, 1717, 2147, 2752, 3290, 4187, 6305, 11051, 12105, 12927, 14357, 14652, 15862, 16172, 17442, 20475, 29330, 30602, 31032, 32782, 38320, 39772, 41750, 45327, 45850, 47437, 51801, 54055, 57526, 61712, 66676, 69230, 70525, 75825, 78547, 83426, 102051, 102835, 108407, 110840, 116622, 117460, 123410, 126876, 130390, 131276, 133952, 164176, 180440, 185680, 186737, 187797, 200751, 201850, 204057, 205165, 206276, 210750, 227955, 229126, 236215, 258130, 261877, 273280, 288862, 298151, 299490, 300832, 308947, 317170, 321322, 326900, 329707, 338200, 373252, 386842, 456780, 470120, 485357, 487065, 525400, 532526, 541501, 548735, 583752, 596926, 612162, 691222, 701442, 709672, 738855, 772927, 777240, 832165, 912210, 942877, 998376, 1000825, 1030447, 1032935, 1035426, 1052947, 1073151, 1075690, 1085876, 1148000, 1150626, 1211852, 1230801, 1263627, 1269140, 1274665, 1288530, 1302470, 1319297, 1341901, 1422527, 1437172, 1443051, 1451892, 1529645, 1553977, 1603217, 1606320, 1634382, 1643790, 1656376, 1662687, 1672176, 1684870, 1697612, 1700805, 1707200, 1720026, 1742587, 1778337, 1781605, 1804565, 1811152, 1817751, 1834301, 1867626, 1874327, 1931770, 1941997, 1959102, 1976282, 1983175, 2021301, 2028272, 2031762, 2066827, 2088010, 2145026, 2239315, 2261362, 2287220, 2343125, 2354387, 2414907, 2495505, 2542155, 2581360, 2589237, 2628802, 2632775, 2700775, 2769642, 2860051, 2876645, 2964551, 3023890, 3075220, 3114001, 3139990, 3179176, 3218605, 3227400, 3231802, 3240615, 3245026, 3267126, 3284860, 3293745, 3320472, 3442080, 3460282, 3478532, 3483102, 3538176, 3542785, 3552012, 3687152, 3758042, 3777060, 3839200, 3872870, 3887345, 3930932, 3950382, 3989426, 4043425, 4077977, 4227722, 4263051, 4323957, 4329052, 4395560, 4467751, 4483297, 4540530, 4703776, 4767742, 4773092, 4783801, 4799887, 4821377, 4837526, 4853702, 5016947, 5060935, 5077480, 5099582, 5138376, 5188470, 5244415, 5250026, 5266877, 5345872, 5396965, 5436872, 5454020, 5465467, 5499880, 5522882, 5551702, 5569030, 5597970, 5667732, 5679401, 5873672, 5891495, 5945126, 5951100, 5969040, 6017012, 6071210, 6077247, 6083287, 6095376, 6186426, 6210820, 6241380, 6253625, 6296577, 6401501, 6432526, 6438740, 6444957, 6501045, 6563650, 6582490, 6613950, 6689760, 6696097, 6708780, 6746901, 6797897, 6984367, 7016772, 7075290, 7088327, 7120972, 7127510, 7193055.

Some larger examples :