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de Polignac n.  and  pernicious n.

First numbers that belong to both sets:

127, 251, 331, 701, 757, 809, 877, 905, 997, 1087, 1199, 1207, 1211, 1243, 1259, 1549, 1597, 1657, 1777, 1807, 1867, 1927, 1969, 2171, 2213, 2231, 2263, 2279, 2293, 2377, 2465, 2503, 2579, 2669, 2789, 2843, 2993, 3119, 3163, 3181, 3187, 3215, 3239, 3299, 3343, 3433, 3505, 3637, 3697, 3779, 3877, 3967, 3985, 4001, 4063, 4153, 4195, 4271, 4311, 4327, 4503, 4633, 4649, 4663, 4717, 4781, 4811, 4813, 4841, 5323, 5405, 5731, 5737, 5761, 5771, 5923, 6021, 6193, 6247, 6403, 6433, 6535, 6539, 6547, 6659, 6673, 6731, 6757, 6791, 6821, 6853, 7151, 7177, 7267, 7297, 7331, 7393, 7431, 7747, 7753, 7807, 7811, 7813, 7841, 8031, 8111, 8257, 8287, 8387, 8399, 8411, 8429, 8467, 8527, 8563, 8621, 8849, 8861, 8915, 8921, 9101, 9307, 9371, 9457, 9517, 9557, 9613, 9641, 9929, 10021, 10121, 10235, 10237, 10253, 10327, 10357, 10391, 10451, 10511, 10513, 10553, 10697, 10781, 10949, 11015, 11081, 11105, 11141, 11231, 11279, 11317, 11335, 11347, 11411, 11533, 11541, 11593, 11743, 11771, 11789, 11801, 11857, 11909, 11921, 12191, 12239, 12251, 12431, 12517, 12595, 12805, 12877, 12929, 12941, 13001, 13091, 13093, 13217, 13351, 13451, 13589, 13603, 13859, 13897, 13961, 14047, 14077, 14143, 14375, 14381, 14407, 14437, 14489, 14531, 14533, 14605, 14611, 14917, 14921, 14975, 14981, 15041, 15071, 15101, 15113, 15121, 15223, 15349, 15359, 15373, 15401, 15521, 15551, 15607, 15853, 15997, 16031, 16109, 16117, 16361, 16507, 16543, 16559, 16629, 16727, 16783, 16867, 17039, 17047, 17077, 17089, 17113, 17137, 17229, 17257, 17305, 17429, 17467, 17489, 17579, 17593, 17669, 17735, 17827, 17861, 17977, 18033, 18089, 18211, 18307, 18427, 18487, 18565, 18611, 18613, 18637, 18787, 18817, 18889, 18897, 18959, 18971, 19057, 19093, 19139, 19273, 19345, 19361, 19483, 20113, 20287, 20383, 20467, 20617, 20623, 20651, 20653, 20693, 20707, 21037.

Some larger examples :