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de Polignac n.  and  evil n.

First numbers that belong to both sets:

149, 337, 373, 509, 599, 907, 977, 1271, 1477, 1529, 1541, 1589, 1619, 1649, 1719, 1829, 1859, 1973, 1985, 2203, 2429, 2683, 2909, 2983, 3029, 3149, 3341, 3353, 3431, 3539, 3643, 3665, 3739, 3817, 3845, 4151, 4229, 4567, 4573, 4589, 4691, 4843, 4889, 5077, 5125, 5143, 5303, 5467, 5557, 5609, 5617, 5729, 5917, 5951, 6001, 6065, 6119, 6161, 6173, 6269, 6283, 6449, 6509, 6521, 6869, 6883, 6941, 7109, 7169, 7199, 7289, 7319, 7379, 7535, 7547, 7583, 7799, 7867, 7901, 7961, 8023, 8087, 8107, 8141, 8159, 8363, 8587, 8669, 8719, 8789, 8873, 8887, 8923, 8929, 8981, 9115, 9239, 9391, 9431, 9473, 9521, 9569, 9581, 9787, 9809, 9907, 9941, 10001, 10007, 10027, 10061, 10079, 10199, 10379, 10409, 10447, 10483, 10753, 10777, 10873, 10963, 11023, 11083, 11137, 11171, 11227, 11239, 11285, 11435, 11437, 11549, 11695, 11729, 12007, 12203, 12223, 12371, 12373, 12401, 12427, 12671, 12749, 12791, 12881, 13083, 13099, 13169, 13393, 13409, 13469, 13619, 13679, 13799, 13841, 13901, 13973, 14021, 14039, 14081, 14101, 14141, 14231, 14279, 14303, 14347, 14383, 14467, 14473, 14681, 15013, 15037, 15043, 15247, 15641, 15719, 15787, 15809, 15869, 16027, 16181, 16213, 16405, 16409, 16499, 16601, 16645, 16739, 16753, 16769, 16787, 16849, 16865, 16973, 17021, 17083, 17269, 17369, 17411, 17437, 17519, 17557, 17651, 17767, 17773, 17849, 17887, 17921, 17957, 18209, 18359, 18391, 18517, 18551, 18607, 18719, 18845, 18869, 18881, 18899, 18911, 19117, 19135, 19177, 19319, 19583, 19649, 19709, 19807, 19889, 19949, 19961, 20143, 20321, 20381, 20437, 20519, 20591, 20599, 20639, 20717, 20737, 20827, 20879, 20893, 20909, 20969, 21073, 21101, 21119, 21163, 21359, 21457, 21463, 21541, 21583, 21637, 21659, 21671, 21683, 21751, 21763, 21793, 21817, 21899, 21907, 22049, 22327, 22331, 22439, 22453, 22769, 22817, 22859, 22919, 22943, 22949, 23111, 23129, 23141, 23179.

Some larger examples :