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cyclic n.  and  de Polignac n.

First numbers that belong to both sets:

1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1529, 1541, 1549, 1589, 1597, 1619, 1649, 1657, 1759, 1777, 1783, 1807, 1829, 1867, 1927, 1969, 1973, 1985, 2171, 2203, 2213, 2231, 2263, 2279, 2293, 2377, 2429, 2465, 2503, 2579, 2669, 2683, 2789, 2843, 2879, 2909, 2983, 2993, 2999, 3029, 3119, 3149, 3163, 3181, 3187, 3215, 3239, 3299, 3341, 3343, 3353, 3431, 3433, 3539, 3637, 3643, 3665, 3697, 3739, 3779, 3817, 3845, 3877, 3967, 3985, 4001, 4013, 4151, 4153, 4195, 4229, 4271, 4327, 4543, 4567, 4573, 4589, 4633, 4649, 4663, 4691, 4717, 4781, 4811, 4813, 4841, 4843, 4889, 5077, 5099, 5143, 5303, 5323, 5557, 5609, 5617, 5729, 5731, 5737, 5761, 5771, 5917, 5923, 5951, 6001, 6065, 6073, 6119, 6161, 6173, 6193, 6247, 6269, 6283, 6403, 6433, 6449, 6463, 6509, 6521, 6535, 6539, 6547, 6637, 6659, 6673, 6731, 6791, 6821, 6869, 6883, 6941, 7109, 7151, 7169, 7177, 7199, 7289, 7297, 7319, 7331, 7343, 7379, 7387, 7393, 7417, 7431, 7517, 7547, 7583, 7603, 7747, 7753, 7783, 7799, 7807, 7811, 7813, 7841, 7867, 7901, 7913, 8023, 8087, 8111, 8159, 8257, 8287, 8363, 8387, 8399, 8411, 8429, 8467, 8527, 8563, 8587, 8621, 8669, 8719, 8789, 8831, 8849, 8861, 8873, 8887, 8915, 8921, 8923, 8929, 8981, 9101, 9115, 9239, 9307, 9371, 9391, 9431, 9473, 9517, 9521, 9557, 9569, 9613, 9787, 9809, 9907, 9929, 9941, 9959, 10001, 10007, 10021, 10027, 10061, 10079, 10199, 10235, 10237, 10253, 10327, 10357, 10379, 10391, 10409, 10447, 10451, 10483, 10511, 10513, 10553, 10607, 10619, 10697, 10753, 10777, 10781, 10873, 10949, 10963, 11015, 11023, 11039, 11069, 11083, 11141, 11171, 11207, 11219, 11227, 11231, 11239, 11279, 11317, 11335.

Some larger examples :