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cyclic n.  and  twin primes

First numbers that belong to both sets:

3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643, 659, 661, 809, 811, 821, 823, 827, 829, 857, 859, 881, 883, 1019, 1021, 1031, 1033, 1049, 1051, 1061, 1063, 1091, 1093, 1151, 1153, 1229, 1231, 1277, 1279, 1289, 1291, 1301, 1303, 1319, 1321, 1427, 1429, 1451, 1453, 1481, 1483, 1487, 1489, 1607, 1609, 1619, 1621, 1667, 1669, 1697, 1699, 1721, 1723, 1787, 1789, 1871, 1873, 1877, 1879, 1931, 1933, 1949, 1951, 1997, 1999, 2027, 2029, 2081, 2083, 2087, 2089, 2111, 2113, 2129, 2131, 2141, 2143, 2237, 2239, 2267, 2269, 2309, 2311, 2339, 2341, 2381, 2383, 2549, 2551, 2591, 2593, 2657, 2659, 2687, 2689, 2711, 2713, 2729, 2731, 2789, 2791, 2801, 2803, 2969, 2971, 2999, 3001, 3119, 3121, 3167, 3169, 3251, 3253, 3257, 3259, 3299, 3301, 3329, 3331, 3359, 3361, 3371, 3373, 3389, 3391, 3461, 3463, 3467, 3469, 3527, 3529, 3539, 3541, 3557, 3559, 3581, 3583, 3671, 3673, 3767, 3769, 3821, 3823, 3851, 3853, 3917, 3919, 3929, 3931, 4001, 4003, 4019, 4021, 4049, 4051, 4091, 4093, 4127, 4129, 4157, 4159, 4217, 4219, 4229, 4231, 4241, 4243, 4259, 4261, 4271, 4273, 4337, 4339, 4421, 4423, 4481, 4483, 4517, 4519, 4547, 4549, 4637, 4639, 4649, 4651, 4721, 4723, 4787, 4789, 4799, 4801, 4931, 4933, 4967, 4969, 5009, 5011, 5021, 5023, 5099, 5101, 5231, 5233, 5279, 5281, 5417, 5419, 5441, 5443, 5477, 5479, 5501, 5503, 5519, 5521, 5639, 5641, 5651, 5653, 5657, 5659, 5741, 5743, 5849.

Some larger examples :