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alternating n.  and  good prime

First numbers that belong to both sets:

5, 29, 41, 67, 101, 127, 149, 307, 347, 541, 563, 569, 587, 727, 929, 967, 2521, 4507, 4783, 4903, 4967, 6521, 6547, 6563, 6761, 6947, 8161, 8167, 8501, 8521, 10589, 14321, 14369, 14387, 16547, 32969, 34123, 36523, 36527, 38149, 38167, 38543, 50741, 50767, 52501, 52529, 52541, 54347, 54361, 54367, 56369, 74143, 74707, 92107, 92143, 92347, 92567, 94307, 98369, 216569, 216703, 216743, 218509, 234121, 234161, 234181, 234187, 238141, 250949, 250963, 254729, 254927, 274783, 278543, 294127, 294149, 294167, 416947, 416989, 492389, 492587, 492967, 494341, 610523, 610541, 614527, 652189, 658547, 672743, 678169, 678329, 678341, 678367, 816763, 818947, 838349, 854921, 874543, 878567, 890707, 890789, 1032341, 1054169, 1072763, 1074509, 1090127, 1090709, 1210369, 1210387, 1210747, 1210787, 1234747, 1236161, 1236383, 1236701, 1238381, 1278107, 1278163, 1278181, 1278323, 1278983, 1294967, 1298909, 1298981, 1418741, 1430587, 1430783, 1470941, 1470947, 1476149, 1476949, 1614149, 1614307, 1652347, 1672301, 1690187, 1690967, 1696943, 1696969, 1818307, 1850749, 1854101, 1854109, 1874987, 3254183, 3490769, 3652543, 3654143, 3654149, 3654349, 3670549, 3670949, 3814301, 3814309, 3872761, 3872767, 5014903, 5014921, 5014927, 5018129, 5050543, 5050567, 5074709, 5074747, 5212703, 5212769, 5212787, 5236909, 5236949, 5410367, 5410723, 5436127, 5436709, 5436721, 5656921, 5658547, 5674909, 5690527, 5838163, 5874709, 5874721, 7014347, 7014367, 7016363, 7016381, 7070143, 7230563, 7232341, 7232363, 7232503, 7232527, 7232543, 7232549, 7236703, 7236767, 7238743, 7238909, 7258747, 7258789, 7258921, 7412381, 7412549, 7474343, 7474361, 7474381, 7834361, 7834583, 7836947, 7836967, 9016901, 9016949, 9018587, 9030347, 9030389, 9030509, 9030709, 9036581, 9052367, 9076183, 9210701, 9216787, 9216943, 9292321, 9292589, 9296981, 9432187, 9434521, 9438563, 9458983, 9612787, 9612989, 9854527, 9854549, 9854563, 9854567, 9854969, 9854981, 9876761, 21232963, 21252107, 21258301, 21258329, 21258527, 21258767, 21272947, 21450103, 21450509, 21450761, 21652783, 21898147, 21898181, 23216363, 23216561, 23250581, 23252183, 23258503, 23258761, 23418509, 23418709, 23430541, 23430949, 23436167, 23436967, 23436989, 23438167, 23438309, 23438507, 23450743, 23450767, 23452147, 23452181, 23452307, 23452901, 23452981, 23452987, 23454703, 23610109, 23650369, 23650789, 23850389.

Some larger examples :