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cyclic n.  and  emirpimes

First numbers that belong to both sets:

15, 51, 85, 115, 123, 143, 159, 177, 185, 187, 221, 265, 319, 321, 329, 335, 339, 341, 413, 415, 437, 493, 511, 533, 551, 553, 559, 581, 589, 623, 629, 667, 685, 771, 781, 817, 835, 851, 871, 893, 899, 913, 923, 933, 951, 985, 1041, 1043, 1057, 1059, 1067, 1079, 1115, 1121, 1135, 1141, 1145, 1149, 1165, 1169, 1177, 1195, 1211, 1219, 1243, 1247, 1253, 1273, 1293, 1313, 1315, 1343, 1347, 1357, 1383, 1387, 1401, 1411, 1415, 1417, 1437, 1469, 1507, 1513, 1527, 1561, 1563, 1577, 1651, 1671, 1679, 1687, 1691, 1707, 1717, 1727, 1735, 1757, 1761, 1765, 1769, 1779, 1795, 1797, 1807, 1817, 1843, 1891, 1897, 1903, 1915, 1923, 1927, 1937, 1961, 1969, 1981, 1985, 2021, 2047, 2051, 2059, 2077, 2219, 2231, 2245, 2257, 2263, 2279, 2413, 2429, 2435, 2443, 2449, 2461, 2479, 2491, 2495, 2603, 2627, 2669, 2807, 2815, 2831, 2839, 2845, 2881, 3007, 3013, 3043, 3047, 3053, 3065, 3071, 3085, 3091, 3093, 3097, 3103, 3131, 3139, 3151, 3173, 3199, 3215, 3227, 3247, 3291, 3309, 3317, 3327, 3365, 3385, 3401, 3415, 3419, 3421, 3431, 3453, 3473, 3503, 3521, 3543, 3563, 3579, 3589, 3599, 3635, 3647, 3651, 3653, 3665, 3667, 3669, 3679, 3687, 3695, 3707, 3713, 3715, 3737, 3743, 3777, 3785, 3809, 3817, 3831, 3839, 3869, 3893, 3899, 3903, 3921, 3949, 3957, 3959, 3979, 3983, 3985, 4109, 4115, 4117, 4121, 4135, 4145, 4151, 4171, 4181, 4183, 4187, 4195, 4307, 4309, 4313, 4321, 4333, 4369, 4385, 4387, 4399, 4511, 4531, 4535, 4537, 4541, 4553, 4571, 4573, 4577, 4579, 4727, 4735, 4765, 4771, 4781, 4963, 4981, 5033, 5053, 5095, 5111, 5123, 5131, 5141, 5143, 5165, 5173, 5177, 5191, 5311, 5317, 5327, 5339, 5345, 5359, 5363, 5371, 5411, 5435, 5447.

Some larger examples :