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interprime n.  and  nude n.

First numbers that belong to both sets:

4, 6, 9, 12, 15, 99, 111, 144, 288, 312, 315, 324, 432, 515, 636, 816, 1113, 1176, 1184, 1197, 1344, 1416, 1488, 1575, 1632, 1715, 1722, 1962, 2112, 2322, 2364, 2444, 2688, 2772, 3168, 3366, 4128, 4224, 4236, 4248, 4332, 4344, 4368, 4444, 4488, 4824, 4896, 4932, 4968, 6126, 6132, 6168, 6384, 6696, 6762, 7224, 8232, 8616, 9135, 9144, 9288, 9315, 9432, 9936, 11115, 11166, 11172, 11226, 11824, 11832, 11916, 12162, 12312, 12336, 12444, 12612, 13155, 13224, 13311, 13932, 14232, 14448, 14824, 15155, 15555, 15575, 16326, 16344, 16362, 16632, 16716, 17248, 18288, 18312, 18864, 19692, 19962, 21216, 21224, 21492, 21612, 21648, 22368, 22444, 22968, 23124, 23292, 23316, 23424, 23616, 23688, 23832, 24144, 24912, 24984, 26112, 26166, 26232, 26262, 26364, 26412, 26712, 27888, 28224, 29196, 29448, 31122, 31131, 31248, 31335, 31368, 31392, 31515, 31848, 31932, 31968, 32166, 32244, 32322, 32412, 32616, 32832, 33288, 33384, 33555, 33768, 33939, 34128, 34212, 34224, 34332, 34344, 34368, 34848, 35535, 35595, 36144, 36648, 36792, 36864, 37317, 38328, 38448, 39195, 39222, 39915, 39933, 41124, 41136, 41224, 41232, 41316, 41412, 41448, 41664, 42184, 42248, 42288, 42672, 42912, 42984, 44184, 44364, 44772, 46224, 46332, 46464, 46824, 46872, 47712, 48144, 48312, 48648, 48864, 49392, 49932, 51135, 51975, 53115, 53595, 53955, 55155, 55335, 61332, 61662, 62124, 62226, 62622, 62928, 63312, 63324, 63648, 63816, 64116, 64164, 64944, 66636, 66648, 68112, 68448, 68664, 68712, 69192, 74172, 76272, 77364, 77931, 79632, 81144, 81168, 81288, 81448, 81624, 81648, 82344, 83232, 84216, 84864, 86112, 88848, 89424, 89496, 91368, 91395, 91791, 92196, 93132, 93492, 93555, 93888, 94896, 95535, 96966, 96984, 98136, 98424, 98928, 99135, 99144, 111124, 111135, 111168, 111222, 111339, 111492, 111515, 111824, 112116, 112244, 112248, 112344, 112362, 112416, 112444, 112632, 113328, 113622, 114128, 114244, 114264.

Some larger examples :