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de Polignac n.  and  lucky n.

First numbers that belong to both sets:

1, 127, 331, 997, 1087, 1597, 1777, 2983, 3163, 3187, 3433, 3697, 4063, 4195, 4567, 4573, 4663, 4717, 4813, 4843, 5617, 5737, 5755, 5923, 6433, 6463, 6535, 6757, 6883, 7405, 7603, 7813, 8107, 8257, 8467, 8719, 9115, 9457, 9613, 9787, 10753, 10873, 11137, 11227, 11239, 11437, 11541, 12007, 12223, 12373, 12427, 12877, 13083, 13285, 13897, 14347, 14407, 14437, 14473, 15121, 15187, 15217, 15247, 16213, 16507, 16867, 17047, 17083, 17229, 17305, 17467, 17767, 17977, 18211, 18787, 18817, 19357, 19807, 21073, 21453, 21751, 22147, 22327, 22507, 22521, 22957, 23173, 23767, 23857, 23893, 24133, 24217, 24523, 24733, 24963, 25057, 25561, 25753, 25909, 25987, 26167, 26413, 27097, 27157, 27475, 27913, 28087, 28483, 28837, 28843, 29317, 29623, 29653, 29689, 29971, 30313, 30409, 30643, 30727, 30793, 31033, 31237, 32407, 32467, 32683, 32713, 33433, 34003, 34171, 34183, 34603, 35221, 35443, 35485, 35917, 36037, 36151, 36187, 36613, 36759, 37087, 37663, 37753, 37963, 38473, 38917, 39007, 39481, 39607, 39643, 40147, 40207, 40323, 40447, 40477, 40923, 40933, 41029, 41107, 41425, 41947, 42169, 42373, 42883, 43207, 43633, 43711, 43873, 43879, 43933, 43957, 44923, 45175, 45277, 45403, 45613, 46129, 46243, 47107, 47677, 47833, 47923, 48115, 48583, 48703, 48847, 48913, 49447, 49585, 49717, 49753, 49807, 50203, 50317, 50407, 50767, 51127, 51283, 51397, 51703, 51793, 51865, 52243, 52357, 52567, 52627, 52807, 52993, 53113, 53491, 53773, 54007, 54223, 54247, 54487, 54517, 54673, 54733, 55423, 55603, 56263, 56289, 56347, 57343, 57607, 57613, 58753, 59233, 59587, 59893, 59917, 60055, 60187, 60403, 60757, 60973, 61237, 62017, 62287, 62323, 63085, 63127, 63223, 63267, 63487, 63583, 64207, 64213, 64357, 64387, 64533, 64647, 64897, 65353, 65503, 65563, 65629, 65713, 65803, 65941, 66427, 66607, 66613, 66667, 66739, 66907, 67015, 67117, 67873, 68053, 68065, 68377, 68467, 68593, 68755, 69073, 69163, 69301, 69583, 69871.

Some larger examples :