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amenable n.  and  de Polignac n.

First numbers that belong to both sets:

1, 149, 337, 373, 509, 701, 757, 809, 877, 905, 977, 997, 1477, 1529, 1541, 1549, 1589, 1597, 1649, 1657, 1777, 1829, 1969, 1973, 1985, 2213, 2293, 2377, 2429, 2465, 2669, 2789, 2909, 2993, 3029, 3149, 3181, 3341, 3353, 3433, 3505, 3637, 3665, 3697, 3817, 3845, 3877, 3985, 4001, 4013, 4153, 4229, 4573, 4589, 4633, 4649, 4717, 4781, 4813, 4841, 4889, 5077, 5125, 5405, 5557, 5609, 5617, 5729, 5737, 5761, 5917, 6001, 6021, 6065, 6073, 6161, 6173, 6193, 6269, 6433, 6449, 6509, 6521, 6637, 6673, 6757, 6821, 6853, 6869, 6941, 7109, 7169, 7177, 7289, 7297, 7389, 7393, 7405, 7417, 7517, 7753, 7813, 7841, 7901, 7913, 7961, 8141, 8257, 8429, 8621, 8669, 8789, 8849, 8861, 8873, 8921, 8929, 8981, 9101, 9457, 9473, 9517, 9521, 9557, 9569, 9581, 9613, 9641, 9809, 9929, 9941, 10001, 10021, 10061, 10121, 10237, 10253, 10357, 10409, 10513, 10553, 10697, 10753, 10777, 10781, 10873, 10949, 11069, 11081, 11105, 11137, 11141, 11285, 11317, 11437, 11533, 11541, 11549, 11593, 11729, 11789, 11801, 11857, 11909, 11921, 11993, 12233, 12373, 12401, 12517, 12733, 12749, 12805, 12877, 12881, 12929, 12941, 13001, 13093, 13169, 13217, 13285, 13297, 13393, 13409, 13469, 13589, 13841, 13897, 13901, 13961, 13973, 14009, 14021, 14077, 14081, 14101, 14141, 14249, 14381, 14437, 14473, 14489, 14533, 14585, 14605, 14681, 14765, 14809, 14917, 14921, 14981, 15013, 15037, 15041, 15101, 15113, 15121, 15149, 15161, 15217, 15349, 15373, 15401, 15521, 15641, 15701, 15809, 15853, 15869, 15957, 15997, 16013, 16025, 16109, 16117, 16165, 16177, 16181, 16213, 16361, 16405, 16409, 16601, 16629, 16645, 16753, 16769, 16849, 16865, 16973, 17021, 17077, 17089, 17113, 17137, 17229, 17257, 17269, 17305, 17369, 17429, 17437, 17489, 17557, 17593, 17669, 17773, 17849, 17861, 17909, 17921, 17957, 17977, 18033, 18089, 18209, 18517, 18565, 18613.

Some larger examples :