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odious n.  and  palprime

First numbers that belong to both sets:

2, 7, 11, 131, 151, 181, 191, 313, 727, 757, 787, 919, 929, 10301, 10501, 10601, 11411, 12421, 12721, 13831, 13931, 14341, 15451, 15551, 16061, 16361, 16561, 16661, 17971, 18481, 19391, 19891, 30403, 30703, 31513, 32323, 32423, 33533, 36263, 36563, 37273, 37573, 38083, 70207, 70507, 71917, 72227, 73037, 74047, 74747, 76367, 76667, 77377, 77477, 77977, 78487, 78787, 79697, 91019, 93139, 94849, 96769, 97579, 98689, 1008001, 1022201, 1028201, 1035301, 1062601, 1065601, 1092901, 1114111, 1117111, 1120211, 1123211, 1129211, 1150511, 1160611, 1177711, 1178711, 1180811, 1186811, 1193911, 1196911, 1212121, 1218121, 1221221, 1245421, 1253521, 1262621, 1278721, 1280821, 1287821, 1328231, 1333331, 1335331, 1338331, 1362631, 1363631, 1374731, 1390931, 1407041, 1422241, 1444441, 1452541, 1463641, 1464641, 1469641, 1486841, 1489841, 1490941, 1520251, 1535351, 1542451, 1550551, 1551551, 1572751, 1579751, 1580851, 1583851, 1589851, 1597951, 1600061, 1630361, 1640461, 1654561, 1657561, 1660661, 1670761, 1688861, 1695961, 1703071, 1707071, 1730371, 1734371, 1748471, 1761671, 1764671, 1793971, 1805081, 1820281, 1823281, 1826281, 1853581, 1856581, 1865681, 1876781, 1879781, 1880881, 1895981, 1917191, 1930391, 1958591, 1963691, 1969691, 1970791, 1976791, 1981891, 1982891, 1984891, 3001003, 3002003, 3016103, 3065603, 3072703, 3155513, 3160613, 3166613, 3181813, 3187813, 3193913, 3198913, 3211123, 3218123, 3223223, 3228223, 3233323, 3236323, 3256523, 3260623, 3283823, 3286823, 3291923, 3305033, 3307033, 3310133, 3315133, 3321233, 3329233, 3337333, 3368633, 3380833, 3391933, 3400043, 3424243, 3439343, 3441443, 3444443, 3447443, 3452543, 3460643, 3479743, 3487843, 3541453, 3558553, 3563653, 3586853, 3594953, 3607063, 3635363, 3646463, 3689863, 3709073, 3716173, 3717173, 3722273, 3728273, 3732373, 3743473, 3746473, 3762673, 3765673, 3768673, 3781873, 3792973, 3799973, 3812183, 3853583, 3867683, 3871783, 3878783, 3913193, 3916193, 3918193, 3927293, 3931393, 3942493, 3948493, 3970793, 3983893, 3997993, 3998993, 7014107, 7046407, 7065607, 7069607, 7073707, 7079707, 7084807, 7093907, 7096907, 7114117, 7129217, 7136317, 7155517, 7158517, 7190917, 7215127, 7226227, 7246427, 7249427, 7257527, 7276727, 7278727, 7302037, 7314137, 7324237, 7327237, 7381837, 7388837, 7403047, 7436347, 7439347, 7452547, 7466647, 7472747, 7475747, 7485847, 7493947, 7507057.

Some larger examples :