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apocalyptic n.  and  de Polignac n.

First numbers that belong to both sets:

251, 977, 1019, 1087, 1199, 1243, 1259, 1271, 1541, 1589, 1777, 1783, 1807, 1829, 1969, 2231, 2279, 2293, 2579, 2669, 2683, 2789, 2843, 2879, 2909, 3029, 3119, 3149, 3181, 3187, 3215, 3239, 3433, 3739, 3967, 4063, 4151, 4195, 4229, 4271, 4327, 4503, 4543, 4567, 4573, 4589, 4633, 4649, 4663, 4717, 4781, 4811, 4843, 4855, 4889, 5077, 5125, 5143, 5303, 5323, 5405, 5467, 5557, 5609, 5729, 5731, 5737, 5761, 5771, 5923, 5951, 6065, 6073, 6173, 6193, 6269, 6283, 6433, 6449, 6463, 6521, 6535, 6547, 6637, 6659, 6673, 6731, 6757, 6853, 6869, 6883, 6941, 7151, 7169, 7177, 7199, 7267, 7289, 7297, 7319, 7331, 7379, 7387, 7389, 7393, 7405, 7417, 7431, 7517, 7535, 7547, 7583, 7603, 7747, 7753, 7783, 7799, 7807, 7811, 7813, 7841, 7867, 7901, 7913, 7961, 8023, 8031, 8087, 8107, 8111, 8141, 8159, 8257, 8287, 8363, 8387, 8399, 8411, 8429, 8467, 8527, 8563, 8587, 8621, 8669, 8789, 8831, 8849, 8861, 8873, 8887, 8915, 8921, 8923, 8981, 9115, 9239, 9307, 9371, 9391, 9431, 9457, 9473, 9517, 9521, 9557, 9581, 9613, 9641, 9787, 9809, 9907, 9929, 9941, 9959, 10001, 10007, 10021, 10027, 10061, 10079, 10121, 10199, 10235, 10237, 10253, 10327, 10379, 10391, 10409, 10447, 10451, 10483, 10511, 10513, 10553, 10607, 10697, 10753, 10777, 10781, 10873, 10949, 10963, 11015, 11023, 11039, 11081, 11083, 11105, 11137, 11141, 11171, 11207, 11219, 11227, 11231, 11239, 11279, 11285, 11317, 11347, 11411, 11435, 11437, 11533, 11541, 11549, 11579, 11593, 11627, 11695, 11729, 11743, 11771, 11789, 11801, 11857, 11909, 11921, 11993, 12007, 12131, 12191, 12203, 12223, 12233, 12239, 12251, 12371, 12373, 12401, 12427, 12431, 12479, 12517, 12595, 12671, 12727, 12731, 12733, 12749, 12791, 12805, 12877, 12881, 12929, 12941, 13001, 13083, 13091, 13093, 13099, 13147, 13169, 13217, 13285, 13297, 13351, 13393.

Some larger examples :