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gapful n.  and  plaindrome

First numbers that belong to both sets:

135, 225, 1111, 1125, 1134, 1136, 1155, 1159, 1188, 1245, 1278, 1335, 1344, 1349, 1368, 1377, 1455, 1456, 1458, 2222, 2225, 2236, 2268, 2349, 2366, 2457, 2688, 3333, 3367, 3456, 3458, 4444, 4455, 4459, 5555, 5568, 6666, 7777, 8888, 9999, 11112, 11115, 11136, 11144, 11145, 11178, 11229, 11235, 11237, 11268, 11355, 11358, 11445, 11448, 11456, 11577, 11799, 12225, 12233, 12255, 12256, 12257, 12258, 12334, 12336, 12345, 12348, 12369, 12555, 12559, 12888, 13335, 13338, 13447, 13455, 13456, 13699, 13788, 13889, 14445, 14459, 14467, 14688, 14778, 15555, 15588, 15599, 15678, 15789, 16668, 16677, 22224, 22225, 22256, 22288, 22333, 22344, 22359, 22568, 23556, 23688, 24466, 24668, 24679, 25677, 25788, 26999, 27888, 33336, 33337, 33355, 33478, 33579, 33668, 34447, 35557, 35568, 36667, 36699, 37777, 44448, 44666, 44688, 45888, 56699, 66999, 111111, 111135, 111136, 111168, 111169, 111225, 111244, 111255, 111258, 111345, 111348, 111359, 111367, 111456, 111555, 111888, 112224, 112245, 112248, 112256, 112333, 112335, 112336, 112338, 112455, 112499, 112557, 112689, 112788, 113344, 113355, 113445, 113449, 113456, 113577, 113688, 113778, 114555, 114588, 114589, 114678, 114779, 115578, 115668, 116679, 122234, 122235, 122238, 122247, 122256, 122336, 122355, 122379, 122444, 122445, 122569, 122688, 122778, 123345, 123456, 123555, 123588, 123678, 123777, 123899, 124455, 124457, 124469, 124488, 124578, 124668, 125568, 125799, 126667, 127889, 133335, 133399, 133455, 133456, 133467, 133488, 133578, 133589, 133668, 133779, 134445, 134478, 134568, 135555, 135558, 135677, 135679, 138888, 144468, 144555, 144558, 144799, 145559, 146699, 146889, 147888, 155567, 156888, 157777, 157788, 159999, 166788, 167778, 167789, 178999, 222222, 222225, 222237, 222348, 222456, 222459, 222488, 222777, 223344, 223366, 223468, 223888, 224448, 224489, 224588, 224666, 224667, 224779, 225568, 226688, 229999, 233479, 233577, 233688, 234668, 235788, 235799, 236669, 237888, 244468, 244499, 244556, 244789, 244888, 245588, 255668, 256679, 256788, 258888, 258999, 277788, 333333, 333445, 333488.

Some larger examples :