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congruent n.  and  de Polignac n.

First numbers that belong to both sets:

127, 149, 373, 509, 599, 701, 757, 877, 905, 959, 997, 1087, 1199, 1207, 1271, 1477, 1541, 1549, 1589, 1597, 1649, 1719, 1759, 1783, 1807, 1829, 1927, 1973, 2213, 2231, 2263, 2279, 2293, 2429, 2465, 2503, 2669, 2789, 2879, 2909, 2983, 2999, 3029, 3119, 3149, 3181, 3215, 3239, 3341, 3343, 3431, 3433, 3637, 3845, 3877, 3967, 3985, 4013, 4063, 4151, 4229, 4271, 4311, 4327, 4503, 4543, 4567, 4573, 4589, 4633, 4649, 4663, 4717, 4781, 4813, 4841, 4843, 4855, 4889, 5077, 5125, 5143, 5303, 5405, 5467, 5557, 5917, 5951, 6001, 6021, 6065, 6073, 6119, 6173, 6247, 6269, 6463, 6509, 6535, 6637, 6757, 6791, 6821, 6853, 6869, 6941, 7109, 7151, 7199, 7289, 7319, 7343, 7389, 7405, 7417, 7431, 7517, 7535, 7583, 7747, 7783, 7799, 7807, 7813, 7901, 8023, 8031, 8087, 8111, 8141, 8159, 8287, 8399, 8429, 8527, 8621, 8669, 8719, 8789, 8831, 8861, 8887, 8981, 9101, 9239, 9391, 9431, 9517, 9521, 9557, 9581, 9613, 9641, 9809, 9941, 9959, 10001, 10007, 10021, 10061, 10079, 10199, 10237, 10253, 10327, 10357, 10391, 10447, 10511, 10553, 10607, 10619, 10777, 10781, 10949, 11015, 11023, 11039, 11069, 11141, 11207, 11231, 11239, 11279, 11285, 11317, 11335, 11437, 11533, 11541, 11549, 11695, 11743, 11789, 11909, 12007, 12191, 12223, 12239, 12373, 12431, 12479, 12517, 12595, 12671, 12727, 12733, 12749, 12791, 12805, 12877, 12929, 12941, 13093, 13169, 13285, 13351, 13469, 13589, 13679, 13735, 13799, 13841, 13901, 13973, 14021, 14023, 14039, 14047, 14077, 14101, 14141, 14143, 14231, 14279, 14303, 14375, 14381, 14383, 14407, 14437, 14531, 14533, 14605, 14639, 14765, 14879, 14917, 14975, 14981, 15013, 15037, 15071, 15101, 15119, 15127, 15149, 15223, 15247, 15349, 15359, 15373, 15551, 15607, 15701, 15719, 15853, 15869, 15943, 15957, 15997, 16013, 16031, 16109, 16117, 16165, 16181, 16213, 16405, 16409.

Some larger examples :