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plaindrome  and  Smith n.

First numbers that belong to both sets:

4, 22, 27, 58, 166, 346, 355, 378, 588, 666, 778, 1111, 1255, 1449, 1678, 2227, 2366, 2556, 2578, 2679, 2688, 2888, 3345, 3366, 4557, 4788, 11358, 11388, 11679, 12226, 12558, 12667, 13369, 13666, 14458, 14566, 14688, 15558, 15778, 16688, 17889, 22234, 22368, 22448, 22558, 22688, 23449, 23566, 23577, 24588, 24678, 26999, 33378, 33669, 34458, 34555, 45679, 111348, 111579, 112368, 112477, 112666, 113456, 113577, 116889, 122377, 122566, 123558, 124558, 126699, 126778, 133377, 133555, 134599, 135558, 135779, 136668, 144679, 144789, 155677, 167888, 188889, 222579, 223357, 223458, 223555, 223557, 224588, 227778, 233558, 234479, 234778, 235678, 244488, 244579, 245578, 255555, 255789, 255888, 256677, 333339, 334678, 344456, 344888, 355888, 356667, 356678, 444489, 444667, 444789, 444888, 445558, 455577, 455667, 555558, 555889, 1112289, 1112499, 1113369, 1122666, 1122799, 1123359, 1123377, 1123555, 1125789, 1133599, 1134688, 1135568, 1135669, 1144777, 1145556, 1145568, 1145569, 1145888, 1177789, 1222222, 1222299, 1222339, 1222566, 1222699, 1223469, 1224499, 1226799, 1227888, 1233789, 1234489, 1234566, 1234669, 1234678, 1235568, 1237779, 1237788, 1244889, 1246778, 1255688, 1255788, 1266666, 1334468, 1334568, 1335666, 1335777, 1344459, 1344669, 1345567, 1345677, 1355678, 1356889, 1377788, 1444446, 1445566, 1457778, 1457779, 1466799, 1555578, 1555788, 1557999, 2222338, 2222358, 2222558, 2223339, 2223346, 2223389, 2225578, 2227999, 2233338, 2233446, 2234569, 2236779, 2245789, 2255888, 2334777, 2335568, 2335666, 2337799, 2344446, 2344558, 2344578, 2345566, 2345899, 2355568, 2367778, 2445678, 2457778, 2556666, 3333488, 3333559, 3334459, 3344559, 3345889, 3356688, 3357789, 3367779, 3444444, 3466779, 4444569, 4445555, 4445599, 4445679, 4448888, 4455567, 4455778, 4458888, 4477779, 4555677, 4555777, 4555779, 4556799, 4666779, 5556777, 6666666, 6666699, 11112666, 11113599, 11114444, 11123779, 11124447, 11124558, 11133888, 11135579, 11136788, 11145559, 11146666, 11155999, 11156678, 11177779, 11222446, 11222779, 11223378, 11224678, 11224777, 11225679, 11225888, 11233338, 11233777, 11235777, 11236689, 11236778, 11236779, 11244469, 11244669, 11244777, 11244779, 11255556, 11255557, 11256666, 11257899, 11277888, 11333388, 11333478, 11333479, 11333588, 11334667, 11335599, 11344666, 11345679, 11345799, 11356688, 11377788, 11444467, 11444468.

Some larger examples :