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de Polignac n.  and  Ulam n.

First numbers that belong to both sets:

1, 905, 1985, 2789, 3029, 3239, 4153, 6283, 7297, 7583, 7603, 8873, 9641, 9809, 9941, 10391, 11219, 11627, 12595, 12749, 13799, 14107, 14303, 14347, 15119, 15121, 16025, 16499, 16543, 16739, 16783, 16849, 17113, 17137, 17411, 17557, 17909, 18427, 19483, 20201, 20323, 20651, 20653, 21037, 21457, 22361, 22439, 22943, 22957, 23099, 23111, 23651, 23935, 24323, 24431, 24433, 24599, 24665, 24697, 25987, 26051, 26615, 27197, 27475, 27673, 27893, 28169, 28499, 29381, 29833, 30131, 30539, 31619, 32257, 33161, 33809, 34525, 34561, 35729, 35947, 36457, 37349, 37391, 37931, 38107, 39277, 39509, 40027, 40049, 40289, 41107, 41591, 42989, 43289, 43565, 43619, 44291, 44609, 44885, 45579, 45581, 45623, 45733, 46033, 46121, 46229, 46363, 46859, 47279, 47455, 48005, 49151, 49447, 49559, 50549, 50617, 50661, 50891, 52103, 52447, 53183, 53227, 53293, 53351, 53549, 53813, 54287, 54329, 54353, 54629, 54827, 55103, 55169, 55409, 56291, 56293, 56767, 56833, 56963, 57119, 57131, 57373, 58067, 58627, 59189, 60271, 60403, 60943, 61109, 61439, 61483, 61649, 61867, 61933, 62407, 62453, 62771, 63247, 63269, 63291, 63487, 64073, 65129, 65363, 65407, 65473, 65713, 65803, 67181, 67379, 67391, 68735, 68755, 69041, 69085, 69239, 69989, 70663, 71755, 72041, 72129, 73495, 73813, 73903, 74069, 74653, 75863, 76691, 76943, 78289, 78959, 79061, 79711, 80383, 81097, 81439, 81505, 83171, 83381, 83533, 83711, 84373, 85199, 85409, 85739, 85927, 86167, 86797, 87083, 87337, 87403, 87469, 87613, 87853, 88327, 88569, 88649, 88781, 88933, 88979, 89497, 89561, 90731, 91393, 91933, 92549, 92737, 95051, 95647, 95977, 96517, 96727, 96903, 97123, 97211, 97289, 97729, 97807, 97961, 98501, 99757, 100087, 100769, 101377, 101551, 101827, 101983, 102071, 102149, 102389, 102719, 103237, 103559, 104077, 104231, 104813, 105289, 105477, 106567, 107647, 107813, 107857, 107923, 108551, 109949, 110213, 110623, 111329, 111395, 111679, 111681, 112453, 112949, 113323, 113411, 113477, 113797, 114941, 115481, 115483.

Some larger examples :