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plaindrome  and  unprimeable n.

First numbers that belong to both sets:

1134, 1135, 1136, 1138, 1144, 1145, 1146, 1148, 1266, 1268, 1334, 1335, 1336, 1338, 1344, 1345, 1346, 1348, 1355, 1356, 1358, 1466, 1468, 1678, 1688, 2366, 2368, 2488, 2566, 2568, 2588, 3478, 3488, 3566, 3568, 3666, 3668, 3688, 3788, 4478, 4555, 4556, 4558, 4578, 4688, 4778, 5678, 7778, 11122, 11124, 11125, 11126, 11128, 11188, 11222, 11224, 11225, 11226, 11228, 11334, 11335, 11336, 11338, 11344, 11345, 11346, 11348, 11378, 11455, 11456, 11458, 11566, 11568, 11666, 11668, 12334, 12335, 12336, 12338, 12355, 12356, 12358, 12366, 12368, 12388, 12444, 12445, 12446, 12448, 12466, 12468, 12666, 12668, 12778, 13344, 13345, 13346, 13348, 13355, 13356, 13358, 13378, 13588, 13778, 14455, 14456, 14458, 14578, 14588, 14678, 15578, 16688, 16778, 17778, 18888, 22234, 22235, 22236, 22238, 22266, 22268, 22334, 22335, 22336, 22338, 22355, 22356, 22358, 22378, 22478, 22555, 22556, 22558, 22588, 22688, 22888, 23344, 23345, 23346, 23348, 23388, 23466, 23468, 23488, 23578, 24455, 24456, 24458, 24566, 24568, 24588, 24666, 24668, 24778, 25555, 25556, 25558, 25688, 25788, 26678, 27788, 28888, 33366, 33368, 33388, 33444, 33445, 33446, 33448, 33555, 33556, 33558, 33666, 33668, 33688, 33788, 34444, 34445, 34446, 34448, 34555, 34556, 34558, 34566, 34568, 34578, 34778, 35555, 35556, 35558, 35588, 35666, 35668, 35688, 35788, 35888, 36666, 36668, 37778, 38888, 44466, 44468, 44478, 44555, 44556, 44558, 44666, 44668, 44678, 45578, 45688, 45788, 46788, 47788, 55555, 55556, 55558, 55566, 55568, 55778, 56888, 66666, 66668, 66678, 66778, 66788, 77888, 111134, 111135, 111136, 111138, 111155, 111156, 111158, 111166, 111168, 111178, 111234, 111235, 111236, 111238, 111244, 111245, 111246, 111248, 111288, 111355, 111356, 111358, 111366, 111368, 111388, 111455, 111456, 111458, 111478, 111555, 111556, 111558, 111566, 111568, 111678, 111688, 111888, 112355, 112356, 112358, 112378, 112388, 112444, 112445, 112446, 112448, 112466, 112468, 112478, 112566, 112568, 112678, 112888.

Some larger examples :