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plaindrome  and  Sophie Germain prime

First numbers that belong to both sets:

2, 3, 5, 11, 23, 29, 89, 113, 179, 233, 239, 359, 1223, 1229, 1289, 1499, 1559, 1889, 2339, 2399, 2459, 2699, 3359, 3389, 3449, 3779, 6899, 11369, 11579, 11699, 12899, 14489, 14669, 14699, 15569, 18899, 22259, 22349, 22469, 23339, 23459, 23669, 25799, 33479, 33569, 35999, 44699, 45569, 45599, 55799, 55889, 58889, 112289, 112349, 112559, 114479, 114599, 115679, 117779, 123449, 133379, 133499, 134489, 134789, 137999, 144569, 144779, 145679, 146669, 157799, 157889, 178889, 188999, 222269, 223589, 224699, 226799, 227789, 244589, 266999, 333479, 333779, 335579, 335669, 335999, 345599, 444569, 445589, 455579, 456899, 556799, 557789, 599999, 788999, 1111289, 1111559, 1112333, 1112339, 1112789, 1117799, 1122389, 1122449, 1122899, 1124699, 1124789, 1125599, 1128899, 1133579, 1136699, 1144589, 1144889, 1145999, 1146779, 1188899, 1223459, 1224599, 1233569, 1233779, 1333679, 1333889, 1336799, 1345559, 1366889, 1446689, 1457999, 1556669, 1678889, 1777799, 2222579, 2223449, 2224559, 2225999, 2227889, 2237789, 2244779, 2244899, 2245679, 2334779, 2344889, 2367899, 2377799, 2444699, 2456999, 2489999, 2557889, 2667779, 3334559, 3336689, 3336779, 3366689, 3377999, 3479999, 3555599, 3566669, 4444469, 4446899, 4556999, 5666789, 6667889, 6689999, 11117789, 11122889, 11125799, 11135669, 11136689, 11137799, 11148899, 11156669, 11188889, 11224799, 11226899, 11234999, 11268899, 11333669, 11344799, 11348999, 11357999, 11455589, 11455799, 11558999, 12222389, 12233579, 12233789, 12237899, 12258899, 12335699, 12337889, 12355559, 12367889, 12449999, 12788999, 13333679, 13344689, 13345679, 13355669, 13357889, 13456799, 13469999, 13555799, 14466899, 14555669, 14557799, 14567999, 15557999, 15677999, 16677779, 22222589, 22226999, 22289999, 22366889, 22458899, 22667999, 22778999, 23333699, 23345579, 23347799, 23355779, 23458889, 23788889, 24445559, 24446699, 24455999, 24466679, 24557999, 24569999, 25577789, 25679999, 28888889, 29999999, 33334799, 33344999, 33357899, 33445589, 33445889, 33567899, 34455689, 34466669, 34558889, 34688999, 35566799, 35668889, 36688889, 44444789, 44555999, 44568899, 45555689, 45558899, 45667889, 45668999, 45688889, 45779999, 46778999, 55555679, 56668889, 66689999, 67889999, 111111113, 111124589, 111133559, 111144899, 111145799, 111222269, 111223379, 111223589, 111255569, 111333899, 111344669, 111355889, 111377789, 111445679, 111488999, 111568889, 112224449, 112235789, 112245779, 112333379, 112334459, 112335689, 112337789, 112338899.

Some larger examples :