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happy n.  and  twin primes

First numbers that belong to both sets:

7, 13, 19, 31, 103, 109, 139, 193, 239, 313, 617, 881, 1033, 1093, 1151, 1277, 1303, 1427, 1481, 1487, 1607, 1697, 1933, 2111, 2309, 2339, 2383, 3001, 3301, 3329, 3331, 3371, 3391, 3463, 3673, 3823, 3931, 4127, 4157, 4217, 4271, 4481, 4517, 4721, 5281, 5417, 5477, 5659, 5741, 5879, 6197, 6361, 6661, 6701, 6703, 6763, 6791, 7127, 7309, 7331, 7457, 7547, 7589, 8233, 8597, 8821, 8969, 9001, 9013, 9857, 10009, 10039, 10067, 10093, 10141, 10303, 10333, 10427, 10457, 11159, 11491, 11719, 11941, 11971, 12823, 13003, 13009, 13339, 13679, 13757, 13903, 13933, 14011, 14081, 14321, 14387, 14449, 14549, 14551, 14561, 15641, 15737, 15887, 16063, 16363, 16451, 16633, 17027, 17191, 17207, 17659, 17911, 18041, 18047, 18253, 18523, 19141, 19421, 19699, 19889, 19963, 20147, 20639, 20717, 20771, 20899, 21011, 21491, 21599, 21841, 22273, 22483, 22573, 22621, 22861, 23039, 23369, 23741, 24107, 24181, 24371, 25801, 25849, 26251, 26711, 27107, 27283, 27739, 27793, 28099, 28351, 28411, 28549, 28621, 29389, 30013, 30091, 30137, 30139, 30269, 30391, 30869, 31247, 31393, 31769, 31771, 31847, 32141, 32369, 32411, 32609, 32803, 33071, 33289, 33589, 33829, 34127, 34157, 34211, 34471, 34499, 35053, 35281, 35839, 36343, 36791, 37571, 37691, 38239, 38329, 38609, 38669, 38749, 38923, 39829, 40127, 41231, 41387, 41761, 42017, 42071, 42181, 42283, 42701, 43781, 43789, 44699, 44701, 44773, 45137, 45317, 47057, 47351, 47389, 47699, 47743, 48121, 48311, 48479, 48731, 49121, 49391, 49549, 49739, 49789, 49937, 51347, 51437, 51769, 52183, 52543, 52861, 53147, 53281, 53717, 54419, 54499, 54583, 55621, 57223, 58231, 58393, 58453, 58907, 59219, 59441, 59671, 60661, 60763, 60899, 60917, 61379, 61471, 63029, 63197, 63199, 63601, 63689, 64151, 64303, 64451, 64663, 65521, 65539, 65719, 67139, 67141, 67211, 67217, 67271, 67411, 67577, 67757, 67931, 68111, 68899, 69193, 69497, 69761, 69931.

Some larger examples :