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de Polignac n.  and  odious n.

First numbers that belong to both sets:

1, 127, 251, 331, 701, 757, 809, 877, 905, 959, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1549, 1597, 1657, 1759, 1777, 1783, 1807, 1867, 1927, 1969, 2171, 2213, 2231, 2263, 2279, 2293, 2377, 2465, 2503, 2579, 2669, 2789, 2843, 2879, 2993, 2999, 3119, 3163, 3181, 3187, 3215, 3239, 3299, 3343, 3433, 3505, 3637, 3697, 3779, 3877, 3967, 3985, 4001, 4013, 4063, 4153, 4195, 4271, 4311, 4327, 4503, 4543, 4633, 4649, 4663, 4717, 4781, 4811, 4813, 4841, 4855, 5099, 5323, 5405, 5731, 5737, 5755, 5761, 5771, 5923, 6021, 6073, 6193, 6247, 6403, 6433, 6463, 6535, 6539, 6547, 6637, 6659, 6673, 6731, 6757, 6791, 6821, 6853, 7151, 7177, 7267, 7297, 7331, 7343, 7387, 7389, 7393, 7405, 7417, 7431, 7517, 7603, 7747, 7753, 7783, 7807, 7811, 7813, 7841, 7913, 8031, 8111, 8257, 8287, 8387, 8399, 8411, 8429, 8467, 8527, 8563, 8621, 8831, 8849, 8861, 8915, 8921, 9101, 9307, 9371, 9457, 9517, 9557, 9613, 9641, 9929, 9959, 10021, 10121, 10235, 10237, 10253, 10327, 10357, 10391, 10451, 10511, 10513, 10553, 10607, 10619, 10697, 10781, 10949, 11015, 11039, 11069, 11081, 11105, 11141, 11207, 11219, 11231, 11279, 11317, 11335, 11347, 11411, 11533, 11541, 11579, 11593, 11627, 11743, 11771, 11789, 11801, 11857, 11909, 11921, 11993, 12131, 12191, 12233, 12239, 12251, 12431, 12479, 12517, 12595, 12727, 12731, 12733, 12805, 12877, 12929, 12941, 13001, 13091, 13093, 13147, 13217, 13285, 13297, 13351, 13451, 13589, 13603, 13735, 13859, 13897, 13961, 14009, 14023, 14047, 14051, 14077, 14107, 14143, 14227, 14249, 14375, 14381, 14407, 14437, 14459, 14489, 14531, 14533, 14585, 14605, 14611, 14639, 14765, 14809, 14879, 14917, 14921, 14975, 14981, 15041, 15059, 15071, 15101, 15113, 15119, 15121, 15127, 15149, 15161, 15187, 15217, 15223, 15299, 15349, 15359, 15373, 15401, 15419, 15521, 15551, 15607.

Some larger examples :