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cyclic n.  and  Honaker prime

First numbers that belong to both sets:

131, 263, 457, 1039, 1049, 1091, 1301, 1361, 1433, 1571, 1913, 1933, 2141, 2221, 2273, 2441, 2591, 2663, 2707, 2719, 2729, 2803, 3067, 3137, 3229, 3433, 3559, 3631, 4091, 4153, 4357, 4397, 4703, 4723, 4903, 5009, 5507, 5701, 5711, 5741, 5801, 5843, 5927, 6301, 6311, 6343, 6353, 6553, 6563, 6653, 6737, 6827, 6971, 7013, 7213, 7283, 7411, 7481, 7523, 7741, 8011, 8821, 9103, 10247, 10429, 10559, 10711, 11071, 11173, 11251, 11353, 11701, 11731, 11821, 12277, 12347, 12613, 13007, 13037, 13127, 13217, 13297, 13327, 13339, 13367, 13417, 13457, 13933, 14159, 14437, 14447, 14843, 14923, 15173, 15193, 15263, 15671, 15761, 15803, 15913, 16073, 16193, 16273, 16451, 16661, 17159, 17189, 17317, 17359, 18013, 18041, 18307, 20533, 20543, 21523, 21617, 22189, 22277, 22613, 22807, 23059, 23531, 24019, 24091, 24733, 24841, 24971, 25219, 25229, 25339, 26119, 26189, 26209, 26309, 26339, 26731, 27059, 27091, 27277, 29023, 29101, 29131, 29243, 29423, 30071, 30091, 30253, 30293, 31177, 31391, 31517, 31567, 32009, 32029, 32059, 32191, 32261, 33493, 33533, 34171, 34259, 34471, 34543, 34703, 35027, 35129, 35149, 35227, 35327, 35537, 35617, 36217, 36307, 36433, 36523, 37019, 37039, 37049, 37117, 37463, 37493, 37717, 37783, 38011, 38711, 40361, 40427, 40627, 40637, 40763, 40823, 41131, 41141, 41333, 41491, 41801, 41851, 41941, 42323, 42373, 42509, 42727, 42737, 43543, 43781, 44221, 44417, 45119, 45181, 45191, 45821, 45833, 46147, 46237, 46723, 47017, 47147, 47293, 47491, 47623, 48109, 48119, 48247, 48463, 48473, 49031, 49411, 49531, 50047, 50119, 50417, 51131, 51203, 51263, 51421, 51473, 52051, 52543, 52553, 52691, 52733, 53309, 53407, 53437, 53591, 53881, 53923, 54013, 54217, 54617, 54727, 54941, 55049, 55207, 55541, 55733, 55823, 56807, 57059, 57089, 57119, 57139, 57271, 57301, 57713, 57943, 58073, 58321, 58411, 58921, 59083, 59093, 59183, 60251, 60601, 61051, 61253, 61703, 62017, 62057.

Some larger examples :