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palindromic n.  and  weak prime

First numbers that belong to both sets:

3, 7, 131, 151, 181, 313, 353, 383, 797, 919, 10501, 10601, 12421, 12721, 13331, 13831, 14741, 15451, 16661, 17471, 19391, 19891, 30203, 30403, 31513, 32323, 33533, 34843, 35053, 35153, 36563, 37573, 38083, 70207, 70507, 71917, 75557, 78487, 90709, 91019, 93139, 93739, 94849, 95959, 96269, 96469, 96769, 97579, 98389, 98689, 1043401, 1062601, 1065601, 1082801, 1160611, 1163611, 1178711, 1180811, 1193911, 1201021, 1212121, 1242421, 1245421, 1250521, 1253521, 1257521, 1276721, 1278721, 1286821, 1287821, 1300031, 1311131, 1317131, 1333331, 1343431, 1362631, 1363631, 1374731, 1407041, 1409041, 1411141, 1422241, 1447441, 1452541, 1456541, 1461641, 1469641, 1489841, 1490941, 1496941, 1508051, 1520251, 1542451, 1550551, 1557551, 1572751, 1580851, 1583851, 1589851, 1597951, 1611161, 1630361, 1658561, 1670761, 1688861, 1703071, 1730371, 1755571, 1764671, 1793971, 1820281, 1831381, 1832381, 1842481, 1883881, 1884881, 1903091, 1908091, 1909091, 1917191, 1924291, 1930391, 1936391, 1952591, 1957591, 1968691, 1970791, 1976791, 1981891, 1982891, 1987891, 1988891, 1995991, 1998991, 3001003, 3002003, 3007003, 3016103, 3026203, 3064603, 3072703, 3073703, 3089803, 3091903, 3095903, 3106013, 3140413, 3158513, 3160613, 3196913, 3212123, 3218123, 3222223, 3223223, 3228223, 3286823, 3291923, 3293923, 3310133, 3315133, 3319133, 3321233, 3329233, 3331333, 3343433, 3353533, 3362633, 3365633, 3411143, 3424243, 3427243, 3439343, 3441443, 3443443, 3449443, 3470743, 3479743, 3485843, 3487843, 3515153, 3517153, 3528253, 3569653, 3589853, 3591953, 3601063, 3618163, 3716173, 3717173, 3728273, 3746473, 3762673, 3763673, 3765673, 3769673, 3774773, 3781873, 3784873, 3793973, 3799973, 3826283, 3858583, 3863683, 3867683, 3871783, 3878783, 3893983, 3899983, 3916193, 3924293, 3938393, 3946493, 3948493, 3970793, 7036307, 7057507, 7065607, 7073707, 7096907, 7115117, 7134317, 7136317, 7145417, 7155517, 7158517, 7177717, 7226227, 7246427, 7250527, 7267627, 7278727, 7310137, 7324237, 7352537, 7354537, 7409047, 7434347, 7436347, 7439347, 7472747, 7493947, 7507057, 7508057, 7521257, 7564657, 7576757, 7594957, 7622267, 7644467, 7662667, 7665667, 7669667, 7693967, 7718177, 7742477, 7747477, 7758577, 7772777, 7791977, 7794977, 7821287, 7832387, 7838387, 7843487, 7943497, 7958597, 7985897, 7996997, 9045409, 9046409, 9067609, 9073709, 9127219, 9128219, 9136319, 9149419, 9179719, 9196919, 9199919, 9200029.

Some larger examples :