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plaindrome  and  twin primes

First numbers that belong to both sets:

3, 5, 7, 11, 13, 17, 19, 29, 59, 137, 139, 149, 179, 199, 227, 229, 239, 269, 347, 349, 569, 599, 1229, 1277, 1279, 1289, 1489, 1667, 1669, 1699, 1789, 1999, 2237, 2239, 2267, 2269, 2339, 2689, 2789, 2999, 3359, 3389, 3467, 3469, 3557, 3559, 4789, 4799, 6689, 6779, 8999, 11117, 11119, 11159, 11489, 11699, 11777, 11779, 12239, 12377, 12379, 13337, 13339, 13399, 13679, 13999, 14447, 14449, 15889, 17789, 22277, 22279, 22367, 22369, 22699, 23369, 23669, 23689, 25577, 25579, 25799, 25999, 26699, 33347, 33349, 33589, 33599, 34469, 34499, 34589, 35899, 36779, 36899, 44699, 45589, 46679, 47777, 47779, 57899, 68899, 78889, 79999, 111119, 111227, 111229, 111269, 111779, 112247, 112249, 112289, 112337, 112339, 113357, 113359, 113777, 113779, 114599, 114689, 114799, 116689, 116789, 122389, 122399, 122599, 123377, 123379, 125789, 125899, 133349, 135599, 144479, 144889, 145679, 155579, 156677, 156679, 156799, 156899, 166667, 166669, 167777, 167779, 168899, 177889, 179999, 222347, 222349, 223337, 223339, 223679, 225779, 233689, 234589, 235789, 235889, 236699, 244667, 244669, 245899, 247999, 255589, 256799, 266689, 266899, 267899, 268999, 277789, 277889, 333449, 334889, 336899, 345599, 345889, 346667, 346669, 357779, 379999, 389999, 448999, 455599, 456899, 467899, 468889, 788999, 1112339, 1112567, 1112569, 1112777, 1112779, 1112899, 1115579, 1116889, 1123349, 1123667, 1123669, 1125557, 1125559, 1125569, 1126667, 1126669, 1128899, 1133357, 1133359, 1133477, 1133479, 1134479, 1134557, 1134559, 1135999, 1136999, 1137889, 1138999, 1145899, 1146779, 1146799, 1155899, 1222229, 1222679, 1223447, 1223449, 1223489, 1223689, 1224479, 1224889, 1225577, 1225579, 1225589, 1225999, 1228889, 1233779, 1234789, 1235789, 1238999, 1245689, 1245779, 1255799, 1266779, 1267789, 1336799, 1344599, 1344779, 1344799, 1344899, 1346999, 1348889, 1378999, 1444789, 1445567, 1445569, 1445669, 1446899, 1447889, 1466999, 1467889, 1477789, 1555999, 1556669, 1577999, 1667777, 1667779, 1667789, 1678889, 2222249, 2223449, 2223467, 2223469, 2223499, 2223677, 2223679, 2224457, 2224459, 2224667, 2224669, 2224679, 2225567, 2225569, 2225579, 2225999, 2233379, 2233499.

Some larger examples :