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happy n.  and  interprime n.

First numbers that belong to both sets:

86, 129, 176, 192, 236, 356, 376, 386, 446, 566, 656, 680, 736, 1003, 1092, 1152, 1158, 1184, 1215, 1281, 1290, 1299, 1390, 1527, 1575, 1581, 1599, 1744, 1785, 1875, 1976, 1995, 2112, 2121, 2157, 2396, 2475, 2554, 2568, 2658, 2685, 2745, 2751, 2811, 2847, 2854, 2933, 3086, 3206, 3213, 3310, 3321, 3465, 3531, 3536, 3620, 3634, 3736, 3879, 3926, 3978, 4096, 4248, 4278, 4449, 4460, 4482, 4555, 4712, 4788, 4807, 4824, 4906, 5133, 5175, 5218, 5306, 5425, 5524, 5544, 5560, 5571, 5714, 5742, 5781, 5817, 5824, 5854, 5997, 6107, 6293, 6320, 6364, 6370, 6376, 6409, 6506, 6566, 6613, 6656, 6798, 6825, 6932, 6987, 7016, 7063, 7106, 7124, 7155, 7245, 7360, 7414, 7425, 7488, 7512, 7630, 7689, 7878, 7930, 8088, 8114, 8175, 8200, 8323, 8517, 8524, 8625, 8679, 8739, 8793, 8877, 8937, 8957, 9004, 9012, 9021, 9046, 9130, 9159, 9167, 9201, 9233, 9302, 9400, 9549, 9594, 9655, 9678, 9758, 9768, 9836, 9915, 9945, 10092, 10122, 10233, 10281, 10332, 10335, 10521, 10578, 10599, 10717, 10785, 10815, 10875, 10976, 11015, 11052, 11058, 11077, 11085, 11118, 11125, 11140, 11154, 11205, 11241, 11267, 11276, 11283, 11314, 11388, 11417, 11538, 11583, 11595, 11627, 11645, 11737, 11804, 11824, 11832, 12009, 12085, 12099, 12108, 12180, 12219, 12279, 12312, 12385, 12411, 12465, 12483, 12507, 12580, 12586, 12628, 12750, 12826, 12835, 12909, 12914, 12921, 13035, 13124, 13185, 13218, 13303, 13320, 13334, 13482, 13518, 13530, 13703, 13754, 13818, 13881, 13998, 14146, 14151, 14156, 14312, 14378, 14404, 14427, 14470, 14511, 14526, 14546, 14562, 14655, 14720, 14734, 14750, 14837, 14889, 14954, 15057, 15075, 15087, 15114, 15138, 15180, 15208, 15270, 15303, 15330, 15375, 15407, 15456, 15464, 15470, 15519, 15546, 15564, 15624, 15642, 15645, 15705, 15735, 15780, 15813, 15820, 15868, 16326, 16336, 16362, 16414.

Some larger examples :