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cyclic n.  and  repunit

First numbers that belong to both sets:

7, 13, 15, 31, 43, 73, 85, 91, 127, 133, 157, 211, 241, 255, 259, 307, 341, 421, 463, 511, 553, 601, 703, 757, 781, 871, 1057, 1093, 1111, 1123, 1261, 1333, 1483, 1561, 1723, 1807, 1885, 1981, 2047, 2071, 2257, 2353, 2551, 2801, 2863, 2955, 2971, 3193, 3307, 3541, 3661, 3907, 4033, 4291, 4369, 4423, 4681, 4831, 5113, 5257, 5461, 5551, 5701, 6007, 6163, 6481, 6643, 6973, 7141, 7483, 7657, 8011, 8191, 8557, 8743, 9313, 9703, 9841, 9901, 10303, 10507, 10921, 11111, 11131, 11155, 11773, 12211, 12433, 12883, 13111, 13807, 14281, 14521, 15007, 15253, 15751, 16003, 16771, 17293, 17557, 18091, 18907, 19183, 19531, 19741, 20023, 20593, 20881, 21757, 21845, 22351, 22621, 22651, 22765, 23257, 23563, 24181, 24493, 25123, 25441, 26083, 26407, 27061, 27931, 28057, 28393, 29071, 29413, 30103, 30451, 30941, 31153, 32221, 32581, 32767, 33307, 33673, 34411, 34783, 35533, 35911, 37057, 37831, 38221, 39403, 40201, 40495, 40603, 41371, 41413, 41821, 42643, 43057, 43891, 44311, 45157, 45583, 46441, 46873, 47743, 47989, 48181, 49507, 50401, 50851, 51757, 52213, 53131, 53593, 54241, 54523, 54991, 55933, 55987, 56407, 57361, 57841, 58807, 59293, 60271, 60763, 61753, 62251, 63253, 64771, 65281, 65535, 65641, 66307, 66823, 67861, 68383, 69961, 71023, 71557, 72631, 73171, 74257, 74803, 75895, 76453, 77563, 78121, 79243, 79807, 81511, 82657, 83233, 84391, 84973, 86143, 86731, 87913, 88507, 88573, 88741, 89701, 90301, 91507, 92113, 93331, 93943, 95173, 95791, 97033, 98911, 99499, 100807, 101443, 102721, 103363, 104653, 106603, 107257, 108571, 109231, 110557, 111151, 111223, 112561, 113233, 114583, 115261, 117307, 118681, 121453, 122851, 123553, 124963, 125671, 127093, 127807, 129241, 131071, 131407, 132133, 133591, 134323, 135793, 136531, 137257, 137561, 138013, 138757, 140251, 141001, 142507, 143263, 145543, 147073, 147841, 149383, 150157, 152491, 154057, 154843, 156421, 157213, 158803, 159601.

Some larger examples :