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de Polignac n.  and  happy n.

First numbers that belong to both sets:

1, 331, 907, 1211, 3029, 3239, 3637, 3643, 3779, 4271, 4633, 4663, 4781, 4811, 4841, 4855, 6065, 6073, 6433, 6463, 6637, 6673, 6791, 7169, 7331, 7379, 7389, 7547, 7603, 7841, 7961, 8141, 8411, 8929, 9307, 9809, 10121, 10447, 10511, 10607, 10697, 11015, 11105, 11541, 11627, 12431, 12671, 12805, 12877, 12941, 13093, 13285, 13393, 13679, 13841, 14047, 14081, 14101, 14231, 14347, 14375, 14381, 14407, 14437, 14459, 14473, 14921, 15101, 15119, 15121, 15299, 15641, 15787, 16181, 16727, 16739, 16769, 16973, 17137, 17229, 17369, 17371, 17411, 17669, 17735, 17827, 18611, 18685, 18899, 19117, 19259, 19889, 20309, 20383, 20639, 20693, 20717, 21101, 21583, 21599, 21671, 21877, 22183, 22327, 22507, 23141, 23257, 23417, 23725, 23893, 24131, 24371, 24683, 24731, 25027, 25057, 25273, 25295, 25453, 25561, 25723, 25777, 25925, 26933, 27161, 27253, 27379, 27611, 28135, 28177, 28273, 28399, 28837, 28865, 29159, 29221, 29297, 29833, 29983, 30109, 30133, 30313, 30319, 30553, 30629, 30643, 30689, 30823, 31033, 31499, 31949, 32009, 32069, 32257, 32309, 32527, 32639, 32741, 32983, 33071, 33209, 33341, 33667, 34181, 34525, 34603, 34721, 34919, 35305, 35485, 35503, 35715, 36035, 36037, 36073, 36209, 36433, 36503, 36613, 36719, 36869, 36971, 37297, 37663, 37691, 37841, 38189, 38417, 38497, 38891, 39007, 39277, 39479, 39491, 40147, 40447, 40477, 40577, 40781, 40811, 40841, 41171, 41347, 41549, 41561, 41939, 42113, 43187, 43417, 43499, 43571, 43603, 43633, 43879, 43919, 43987, 44077, 44107, 44255, 44363, 44429, 44489, 44543, 44849, 45161, 45423, 45583, 46033, 46425, 46663, 47041, 47231, 47489, 47699, 47857, 47939, 48101, 48131, 48371, 48701, 48731, 48757, 48973, 49319, 49391, 49451, 49469, 49739, 49783, 49837, 49897, 50471, 50741, 50879, 51011, 51027, 51119, 51283, 51461, 51697, 51823, 52273, 52529, 52567, 52651, 53147, 53227, 53503, 53545, 53755, 54161, 54491, 54707, 54829, 55423, 55435, 55483, 55553, 55805, 56197.

Some larger examples :