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happy n.  and  Ulam n.

First numbers that belong to both sets:

1, 13, 28, 82, 97, 219, 236, 319, 356, 409, 566, 673, 739, 820, 847, 1030, 1125, 1257, 1335, 1472, 1724, 1748, 1812, 1814, 1900, 1902, 2093, 2112, 2115, 2254, 2393, 2484, 2511, 2572, 2574, 2581, 2584, 2630, 2748, 2811, 2833, 2899, 2919, 3029, 3031, 3038, 3065, 3068, 3092, 3109, 3153, 3239, 3290, 3371, 3456, 3605, 3608, 3698, 3806, 3962, 4040, 4118, 4148, 4187, 4258, 4363, 4365, 4517, 4536, 4878, 4900, 5118, 5181, 5272, 5274, 5335, 5533, 5812, 5826, 6049, 6107, 6239, 6308, 6354, 6435, 6550, 6938, 7063, 7158, 7217, 7366, 7522, 7581, 7603, 7691, 7754, 7884, 7930, 8018, 8060, 8099, 8128, 8424, 8487, 8512, 8639, 8751, 8812, 8969, 9013, 9120, 9313, 9377, 9809, 9836, 9980, 10088, 10090, 10112, 10222, 10247, 10288, 10366, 10393, 10457, 10547, 10679, 11085, 11102, 11168, 11195, 11197, 11229, 11234, 11280, 11388, 11432, 11456, 11627, 11647, 11737, 12033, 12057, 12074, 12143, 12267, 12292, 12438, 12465, 12470, 12526, 12570, 12580, 12622, 12658, 12705, 12707, 12810, 12856, 12900, 12939, 13030, 13032, 13066, 13281, 13303, 13330, 13411, 13457, 14044, 14129, 14281, 14283, 14327, 14347, 14408, 14427, 14449, 14735, 14801, 14840, 14945, 15033, 15082, 15119, 15121, 15138, 15141, 15280, 15622, 15642, 15644, 15688, 15705, 15788, 15820, 15832, 16245, 16306, 16336, 16397, 16399, 16597, 16700, 16722, 16739, 16741, 16937, 16939, 17069, 17071, 17137, 17240, 17303, 17330, 17345, 17411, 17543, 17587, 17758, 17782, 18014, 18044, 18210, 18381, 18432, 18513, 18625, 18750, 18772, 19033, 19590, 19629, 19676, 19698, 19722, 19766, 19893, 20157, 20177, 20199, 20257, 20260, 20455, 20851, 20910, 20998, 21015, 21047, 21105, 21108, 21110, 21132, 21149, 21220, 21279, 21347, 21411, 21413, 21538, 21562, 21628, 21645, 21750, 21865, 21990, 22012, 22075, 22249, 22344, 22378, 22381, 22442, 22483, 22508, 22676, 22723, 22725, 22791, 22816.

Some larger examples :