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de Polignac n.  and  interprime n.

First numbers that belong to both sets:

1207, 1243, 3353, 3505, 3665, 7747, 7913, 9569, 11015, 11627, 12131, 12233, 12427, 12595, 16753, 17437, 18487, 18685, 19807, 20737, 21667, 21793, 22521, 23503, 24323, 24665, 25627, 25753, 25925, 26303, 26567, 26675, 26807, 27097, 27161, 28137, 29971, 30415, 30823, 32347, 33397, 33917, 35917, 35957, 36577, 39307, 40105, 40477, 40745, 42095, 42343, 43529, 45101, 45847, 46193, 46425, 47345, 48085, 49319, 49837, 50479, 50661, 51223, 51395, 51557, 52243, 53447, 53491, 54127, 54157, 60059, 61769, 63403, 64533, 64647, 65593, 68065, 70723, 71017, 71243, 72325, 73777, 75187, 76931, 77767, 78329, 78959, 79895, 80045, 80257, 80903, 81337, 83863, 85871, 86097, 86653, 87371, 88013, 88123, 89437, 89879, 91417, 92773, 92815, 93839, 94241, 95183, 96007, 96205, 97139, 97271, 97535, 97631, 98029, 99161, 99673, 100031, 100285, 100453, 101039, 101353, 101495, 102865, 103539, 103693, 103957, 104465, 105823, 105877, 106267, 106567, 106943, 107083, 107767, 108547, 108733, 109813, 111201, 112793, 112837, 113507, 113579, 113803, 114101, 114305, 115039, 116417, 116519, 117401, 118511, 118577, 118825, 118949, 120187, 120613, 123537, 124483, 125377, 125533, 125875, 126217, 126427, 126697, 127895, 128643, 130063, 130177, 130963, 131339, 131531, 131737, 133001, 133163, 133207, 133397, 133487, 133751, 134765, 135125, 136151, 137797, 138065, 139073, 139151, 139807, 141569, 141607, 142685, 142745, 143023, 143431, 144213, 144433, 145583, 146129, 146167, 146353, 146789, 148507, 148841, 149591, 149723, 149797, 150875, 150973, 151543, 151723, 153253, 155467, 155563, 156697, 156893, 158095, 160597, 162131, 162215, 163205, 163451, 163747, 164161, 164285, 165337, 166637, 168725, 169295, 172015, 173623, 173833, 173951, 174209, 174643, 175487, 176141, 176407, 177817, 179873, 182221, 182555, 182975, 183197, 183367, 184265, 187957, 188379, 189571, 189907, 192997, 193481, 194905, 195247, 195463, 195703, 197227, 197353, 197493, 197915, 198103, 199229, 199541, 200751, 201145, 201815, 202539, 202673, 202723, 203737, 206227, 206495, 206945, 209731, 209755, 211481, 212233, 212975, 214357, 215933, 218605, 218803, 220777, 221523.

Some larger examples :