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happy n.  and  pancake n.

First numbers that belong to both sets:

1, 7, 79, 301, 326, 379, 904, 1177, 1771, 2212, 2557, 2851, 4096, 4187, 4852, 5254, 6329, 6904, 8002, 9046, 12562, 15226, 17956, 22156, 23006, 23872, 24311, 25652, 26336, 26797, 27262, 27967, 30382, 30629, 33931, 34454, 36316, 38227, 38782, 40187, 40471, 41617, 44552, 45151, 45452, 47587, 50722, 51682, 52327, 52651, 55612, 56281, 58312, 60032, 68266, 73154, 81407, 82216, 85079, 85492, 89254, 92666, 98347, 101476, 102832, 104654, 105112, 116404, 117856, 120787, 126254, 129287, 140716, 143381, 146071, 146612, 148786, 150976, 153736, 161597, 171406, 189421, 198136, 199397, 203842, 212879, 215497, 218792, 226129, 229504, 230182, 235642, 238396, 241166, 243952, 262451, 266086, 271954, 287662, 292231, 296066, 296836, 297607, 303032, 312446, 322807, 324416, 330892, 333337, 344036, 361676, 363379, 368512, 372817, 379757, 381502, 385882, 414506, 424582, 425504, 429202, 431057, 449827, 472879, 473852, 486592, 493522, 498502, 502504, 504511, 506522, 529936, 535096, 539242, 542362, 543404, 548629, 550726, 560212, 570847, 571916, 580504, 586987, 588071, 590242, 592417, 593506, 595687, 604451, 619942, 626641, 628882, 631127, 632251, 660676, 692077, 715807, 729029, 738721, 755836, 781876, 783127, 786886, 793171, 805816, 823687, 832696, 839161, 845651, 846952, 850861, 853472, 869222, 879802, 887779, 909227, 910576, 913277, 914629, 943252, 955654, 970922, 973711, 984907, 986311, 987716, 991937, 998992, 1004654, 1021736, 1024597, 1034642, 1040404, 1050526, 1051976, 1054879, 1060697, 1081186, 1084129, 1085602, 1134772, 1143829, 1162051, 1165102, 1183492, 1194286, 1202026, 1244254, 1268029, 1295246, 1306537, 1309772, 1311391, 1332529, 1337431, 1340704, 1376971, 1407004, 1442452, 1454366, 1471471, 1492129, 1497316, 1514671, 1518154, 1519897, 1530376, 1540891, 1569107, 1581532, 1601156, 1617302, 1646206, 1675366, 1695562, 1701091, 1710326, 1745647, 1785106, 1790779, 1809754, 1825006, 1826917, 1836487, 1873081, 1880831, 1890541, 1902226, 1917862, 1925704, 1935529, 1945379, 1967137, 1987022, 2021056, 2051326, 2055379, 2061466, 2069596, 2071631, 2085904, 2087947, 2100226, 2126954, 2153851, 2160082, 2166322, 2172571, 2174656, 2206051, 2222887, 2224996, 2248261, 2261002, 2263129, 2290871, 2293012, 2299441, 2301586, 2308027.

Some larger examples :