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cyclic n.  and  Sophie Germain prime

First numbers that belong to both sets:

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003, 2039, 2063, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2459, 2543, 2549, 2693, 2699, 2741, 2753, 2819, 2903, 2939, 2963, 2969, 3023, 3299, 3329, 3359, 3389, 3413, 3449, 3491, 3539, 3593, 3623, 3761, 3779, 3803, 3821, 3851, 3863, 3911, 4019, 4073, 4211, 4271, 4349, 4373, 4391, 4409, 4481, 4733, 4793, 4871, 4919, 4943, 5003, 5039, 5051, 5081, 5171, 5231, 5279, 5303, 5333, 5399, 5441, 5501, 5639, 5711, 5741, 5849, 5903, 6053, 6101, 6113, 6131, 6173, 6263, 6269, 6323, 6329, 6449, 6491, 6521, 6551, 6563, 6581, 6761, 6899, 6983, 7043, 7079, 7103, 7121, 7151, 7193, 7211, 7349, 7433, 7541, 7643, 7649, 7691, 7823, 7841, 7883, 7901, 8069, 8093, 8111, 8243, 8273, 8513, 8663, 8693, 8741, 8951, 8969, 9029, 9059, 9221, 9293, 9371, 9419, 9473, 9479, 9539, 9629, 9689, 9791, 10061, 10091, 10163, 10253, 10271, 10313, 10331, 10529, 10589, 10613, 10691, 10709, 10733, 10781, 10799, 10883, 11171, 11321, 11369, 11393, 11471, 11519, 11549, 11579, 11699, 11783, 11801, 11813, 11831, 11909, 11939, 12011, 12041, 12101, 12119, 12203, 12263, 12329, 12653, 12671, 12791, 12821, 12899, 12923, 12959, 13001, 13049, 13229, 13313, 13451, 13463, 13553, 13619, 13649, 13763, 13883, 13901, 13913, 14009, 14081, 14153, 14159, 14249, 14303, 14321, 14489, 14561, 14621, 14669, 14699, 14741, 14783, 14831, 14879, 14939, 15101, 15161, 15173, 15233, 15269, 15401, 15569, 15629, 15773, 15791, 15803, 15923, 16001, 16091, 16253.

Some larger examples :