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plaindrome  and  weak prime

First numbers that belong to both sets:

3, 7, 13, 19, 23, 47, 89, 113, 139, 167, 199, 229, 233, 337, 349, 359, 389, 449, 467, 577, 677, 1129, 1237, 1259, 1279, 1459, 1489, 1499, 1559, 1669, 1699, 1789, 1889, 1999, 2239, 2269, 2357, 2399, 2447, 2477, 2557, 2689, 2699, 2777, 3347, 3469, 3499, 3559, 3677, 3779, 3889, 4567, 4679, 4789, 4889, 5669, 6679, 11119, 11177, 11279, 11399, 11447, 11689, 11779, 11789, 12269, 12289, 12347, 12379, 12457, 12479, 12589, 12799, 12899, 13339, 13399, 13469, 13477, 13499, 13577, 13789, 13999, 14449, 14489, 14669, 15559, 15569, 15889, 16889, 22279, 22349, 22369, 22679, 22699, 23339, 23459, 23567, 23677, 23689, 23899, 24799, 24889, 25579, 25589, 25679, 25999, 27779, 33349, 33359, 33469, 33569, 33589, 34679, 35677, 35899, 35999, 44777, 45557, 45569, 45589, 45599, 45677, 45779, 47779, 48889, 49999, 55589, 55667, 55799, 56999, 66889, 67789, 68899, 77899, 78889, 79999, 111127, 111149, 111229, 111347, 111599, 111667, 111799, 112223, 112249, 112339, 112349, 112577, 112589, 113359, 113467, 113567, 113779, 114479, 114577, 114679, 114799, 114889, 116689, 117779, 117889, 117899, 122267, 122279, 122299, 122389, 122477, 122599, 122789, 123379, 123457, 123667, 123677, 124477, 124567, 124577, 124679, 124799, 125669, 125899, 133337, 133499, 133999, 134789, 135799, 135899, 136777, 136889, 136999, 137777, 137999, 138899, 139999, 144569, 144779, 144889, 144899, 145777, 146777, 147779, 147799, 155599, 155699, 155777, 156679, 156799, 157799, 166669, 166679, 167779, 167899, 177889, 222337, 222349, 222367, 222389, 222499, 222557, 222679, 222779, 222799, 223339, 223367, 223469, 223589, 224449, 224569, 224579, 225689, 225889, 226669, 226789, 226799, 233357, 233489, 233557, 233599, 233669, 233689, 233777, 234467, 234589, 234599, 235559, 235679, 235699, 235789, 244567, 244589, 244669, 245789, 245899, 246689, 247889, 247999, 255589, 255667, 255679, 256699, 266689, 266899, 266999, 268999, 277789, 277999, 333349, 333457, 333589, 333679, 334447, 334667, 334699, 335567, 335669, 336689, 338999, 345679, 345689, 345889, 346669.

Some larger examples :