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happy n.  and  prime n.

First numbers that belong to both sets:

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093, 1151, 1277, 1303, 1373, 1427, 1447, 1481, 1487, 1511, 1607, 1663, 1697, 1733, 1847, 1933, 2003, 2039, 2063, 2111, 2221, 2309, 2333, 2339, 2383, 2393, 2417, 2557, 2693, 2741, 2833, 2851, 2903, 2963, 3001, 3019, 3067, 3079, 3083, 3109, 3137, 3209, 3301, 3313, 3319, 3323, 3329, 3331, 3371, 3391, 3463, 3607, 3637, 3643, 3673, 3709, 3779, 3797, 3803, 3823, 3833, 3907, 3923, 3931, 4111, 4127, 4157, 4217, 4271, 4363, 4441, 4447, 4481, 4517, 4663, 4721, 4751, 4817, 4871, 5147, 5227, 5281, 5417, 5471, 5477, 5527, 5569, 5659, 5741, 5821, 5879, 5897, 5987, 6037, 6053, 6073, 6163, 6197, 6203, 6329, 6337, 6343, 6353, 6361, 6367, 6373, 6389, 6637, 6661, 6673, 6701, 6703, 6719, 6733, 6763, 6791, 6803, 6899, 6917, 6971, 6983, 7039, 7127, 7309, 7331, 7451, 7457, 7481, 7541, 7547, 7589, 7603, 7691, 7793, 7841, 7937, 8081, 8147, 8233, 8369, 8521, 8597, 8693, 8699, 8741, 8821, 8929, 8963, 8969, 9001, 9007, 9013, 9103, 9133, 9203, 9323, 9377, 9587, 9623, 9689, 9829, 9857, 10009, 10039, 10067, 10093, 10141, 10151, 10177, 10211, 10247, 10303, 10333, 10337, 10427, 10457, 10487, 10607, 10663, 10733, 10771, 10847, 10903, 11113, 11131, 11149, 11159, 11197, 11243, 11251, 11311, 11423, 11467, 11471, 11483, 11491, 11519, 11681, 11719, 11941, 11971, 12011, 12101, 12143, 12149, 12347, 12413, 12437, 12473, 12491, 12511, 12583, 12671, 12743, 12823, 12841, 12853, 12941, 12959, 13003, 13009, 13033, 13037, 13093, 13177, 13241, 13309, 13339, 13411, 13421, 13457, 13487, 13499, 13577, 13679, 13697, 13757, 13841, 13903, 13933, 13967, 14011.

Some larger examples :