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emirpimes  and  plaindrome

First numbers that belong to both sets:

15, 26, 39, 49, 58, 115, 122, 123, 129, 155, 158, 159, 169, 177, 178, 226, 289, 335, 339, 355, 559, 589, 667, 899, 1115, 1119, 1135, 1137, 1145, 1149, 1169, 1177, 1234, 1238, 1247, 1299, 1346, 1347, 1357, 1366, 1379, 1466, 1469, 1569, 1577, 1679, 1779, 2234, 2245, 2257, 2258, 2279, 2446, 2449, 2458, 2479, 2669, 3334, 3338, 3446, 3455, 3459, 3579, 3589, 3599, 3667, 3669, 3679, 3777, 3778, 3899, 4555, 4577, 4579, 5567, 5579, 5677, 5699, 5899, 5999, 6667, 6889, 7778, 7779, 7799, 7899, 7999, 11146, 11147, 11157, 11179, 11189, 11199, 11237, 11246, 11249, 11267, 11335, 11357, 11366, 11378, 11389, 11449, 11459, 11477, 11479, 11558, 11566, 11569, 11599, 11667, 11669, 11678, 11899, 12223, 12226, 12233, 12235, 12237, 12335, 12346, 12359, 12367, 12399, 12458, 12469, 12559, 12599, 12667, 12679, 12779, 13346, 13378, 13379, 13449, 13459, 13555, 13579, 13589, 13666, 14458, 14569, 14579, 14666, 14789, 14799, 14999, 15578, 15677, 15779, 15899, 15999, 16667, 16679, 16689, 17777, 18889, 22223, 22234, 22238, 22249, 22255, 22333, 22459, 22478, 22489, 22499, 22579, 22889, 24449, 24478, 24569, 24667, 24779, 24899, 26689, 28999, 33357, 33379, 33389, 33445, 33455, 33458, 33499, 33559, 33667, 33689, 33778, 33779, 34449, 34477, 34555, 34559, 34579, 34789, 34889, 34999, 35567, 35599, 35669, 35689, 35699, 35779, 36667, 36679, 36889, 38999, 45559, 47899, 55577, 55679, 55789, 55999, 56669, 56789, 56899, 58999, 66667, 66779, 68999, 78899, 111139, 111158, 111169, 111178, 111235, 111238, 111277, 111346, 111349, 111369, 111379, 111449, 111455, 111466, 111499, 111557, 111589, 111669, 112226, 112235, 112333, 112345, 112367, 112389, 112469, 112477, 112478, 112555, 112579, 112777, 113389, 113447, 113455, 113459, 113569, 113579, 113599, 113669, 113689, 114447, 114449, 114459, 114557, 114559, 114566, 114567, 114667, 114677, 115567, 115579, 115678, 115789, 115999, 116777, 117778, 118889, 118999, 119999, 122257.

Some larger examples :