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cyclic n.  and  strong prime

First numbers that belong to both sets:

11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 569, 587, 599, 613, 617, 631, 641, 659, 673, 701, 719, 727, 739, 751, 757, 769, 787, 809, 821, 827, 853, 857, 877, 881, 907, 929, 937, 967, 991, 1009, 1019, 1031, 1049, 1061, 1087, 1091, 1117, 1151, 1163, 1181, 1213, 1229, 1249, 1277, 1289, 1297, 1301, 1319, 1361, 1399, 1423, 1427, 1447, 1451, 1471, 1481, 1487, 1523, 1543, 1549, 1567, 1579, 1597, 1607, 1619, 1657, 1663, 1667, 1693, 1697, 1721, 1733, 1741, 1777, 1783, 1787, 1801, 1823, 1847, 1861, 1867, 1871, 1877, 1901, 1931, 1949, 1973, 1987, 1993, 1997, 2011, 2027, 2053, 2063, 2081, 2087, 2111, 2129, 2137, 2141, 2153, 2203, 2237, 2267, 2281, 2293, 2309, 2333, 2339, 2347, 2371, 2377, 2381, 2389, 2411, 2437, 2459, 2467, 2473, 2503, 2521, 2531, 2539, 2549, 2579, 2591, 2609, 2617, 2647, 2657, 2671, 2683, 2687, 2707, 2711, 2729, 2741, 2749, 2767, 2789, 2797, 2801, 2819, 2833, 2851, 2857, 2879, 2897, 2953, 2969, 2999, 3011, 3019, 3037, 3061, 3079, 3109, 3119, 3163, 3167, 3181, 3187, 3203, 3217, 3251, 3257, 3299, 3319, 3329, 3343, 3359, 3371, 3389, 3407, 3433, 3449, 3457, 3461, 3467, 3491, 3511, 3527, 3539, 3557, 3571, 3581, 3607, 3613, 3631, 3659, 3671, 3691, 3697, 3719, 3727, 3761, 3767, 3793, 3821, 3847, 3851, 3877, 3907, 3917, 3929, 3943, 3989, 4001, 4019, 4049, 4073, 4091, 4127, 4153, 4157, 4201, 4211, 4217, 4229, 4241, 4253, 4259, 4271, 4283, 4327, 4337, 4349, 4357, 4391, 4421, 4441, 4447, 4481, 4507, 4513, 4517.

Some larger examples :