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happy n.  and  pronic n.

First numbers that belong to both sets:

1122, 1332, 5112, 6320, 7482, 8742, 9120, 10920, 11342, 12210, 15750, 17292, 20306, 28056, 36290, 37442, 43056, 43472, 44732, 45156, 53130, 54990, 56882, 59780, 65280, 74256, 74802, 76452, 79806, 81510, 92112, 95790, 99540, 104652, 108570, 113232, 116622, 123552, 124256, 127092, 129960, 132132, 138012, 142506, 143262, 154056, 156420, 163620, 165242, 166872, 167690, 181050, 187922, 196692, 204756, 207480, 209306, 211140, 223256, 224202, 227052, 228962, 234740, 244530, 246512, 247506, 252506, 253512, 265740, 268842, 275100, 281430, 297570, 306362, 311922, 317532, 325470, 331200, 338142, 340472, 342810, 347510, 353430, 364212, 427062, 437582, 440232, 441560, 448230, 456300, 474032, 493506, 513372, 516242, 520562, 543906, 548340, 552792, 560252, 572292, 576840, 612306, 615440, 621732, 634412, 639200, 647220, 650442, 661782, 679800, 684756, 689730, 694722, 709806, 745632, 771762, 777042, 778806, 819930, 863970, 886422, 899652, 909162, 918722, 930260, 932190, 949650, 955506, 973182, 977132, 991020, 993012, 1009020, 1013042, 1023132, 1029210, 1031240, 1043462, 1045506, 1072260, 1101450, 1124660, 1143830, 1167480, 1182656, 1200120, 1202312, 1204506, 1206702, 1213302, 1217712, 1222130, 1230990, 1233210, 1235432, 1264500, 1309880, 1321350, 1335180, 1360722, 1381800, 1391220, 1419672, 1422056, 1426830, 1431612, 1436402, 1443602, 1458056, 1482306, 1528932, 1533882, 1546292, 1548780, 1558752, 1568756, 1609092, 1624350, 1629452, 1632006, 1637120, 1657656, 1665390, 1712172, 1730540, 1754300, 1764912, 1786232, 1823850, 1845522, 1853682, 1867322, 1889250, 1911306, 1922382, 1947420, 1967006, 1978242, 1981056, 2006472, 2009306, 2026352, 2043470, 2052056, 2057790, 2069282, 2072160, 2075040, 2086580, 2101050, 2112662, 2124306, 2127222, 2130140, 2135982, 2156492, 2162370, 2203740, 2206710, 2209682, 2236520, 2263520, 2290682, 2296740, 2311920, 2379306, 2397852, 2410256, 2413362, 2432040, 2435160, 2510640, 2552006, 2613072, 2619542, 2639000, 2648756, 2678132, 2701092, 2757260, 2780556, 2790570, 2817362, 2837540, 2847656, 2984256, 3022382, 3032822, 3124056, 3188010, 3223820, 3241800, 3256220, 3259830, 3263442, 3277910, 3336102, 3372732, 3383760, 3391122, 3402180, 3413256, 3424350, 3502512, 3517500, 3540042, 3555110, 3619506, 3627120, 3634742, 3676806, 3730692, 3761660, 3804450, 3859260, 3867122, 3902600, 4030056, 4034072, 4046132.

Some larger examples :