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equidigital n.  and  nude n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 11, 15, 111, 112, 115, 122, 135, 155, 162, 175, 224, 384, 448, 1113, 1115, 1184, 1296, 1335, 1344, 1424, 1555, 1888, 1944, 2144, 2688, 2744, 2848, 2916, 3888, 4288, 6144, 6912, 11111, 11122, 11136, 11155, 11313, 11328, 11355, 11488, 11515, 11535, 11555, 11711, 11824, 11872, 11888, 12112, 12128, 12224, 12448, 12864, 13155, 13377, 13632, 13713, 13797, 14128, 15115, 15155, 15575, 15715, 16128, 16464, 17717, 18112, 18144, 18432, 18624, 18816, 19926, 21184, 21728, 22112, 22144, 22326, 23424, 24192, 24224, 24448, 26112, 27216, 28416, 31488, 31995, 33135, 33939, 36288, 36864, 37632, 39375, 42112, 44224, 44288, 46848, 47488, 48128, 48384, 48448, 59535, 61236, 81648, 82944, 84224, 88448, 96768, 111115, 111132, 111184, 111424, 111488, 111515, 111915, 112266, 112448, 112896, 113115, 113133, 113319, 113535, 113664, 113913, 114112, 114432, 114696, 114848, 115155, 117775, 117824, 118112, 118144, 118272, 118288, 118464, 118848, 119232, 121184, 121222, 121424, 121662, 121824, 122112, 122224, 122288, 122688, 123648, 124112, 124144, 124224, 124448, 126144, 126336, 128144, 128224, 128848, 131115, 131133, 131136, 131139, 131328, 131331, 131424, 132192, 132864, 133113, 133191, 133248, 133326, 133632, 134136, 134832, 136128, 138624, 139131, 141184, 141248, 141312, 141424, 141696, 141888, 141912, 142112, 142144, 142288, 142464, 142848, 143232, 143424, 144112, 144224, 144384, 144448, 146112, 146688, 148112, 148176, 148224, 148288, 148416, 148448, 151115, 155115, 155555, 161664, 162432, 162816, 164616, 166212, 168192, 168384, 169128, 169344, 169362, 171115, 172872, 173733, 174272, 179739, 181248, 181824, 181888, 182144, 182784, 182848, 183168, 186432, 187712, 188128, 188272, 188288, 188928, 189216, 191133, 191232, 193995, 196992, 198144, 198288, 211288, 211424, 211722, 212128, 213888, 214112, 214144, 218112, 218848, 221248, 221616, 221824, 221888, 222264, 222848, 222912, 223488, 226962, 228128, 228224, 228864, 229392, 231168, 238464, 238848, 239112, 241184, 241472, 241888, 242688, 242848, 243648, 244224, 244448, 246144, 247744, 248128.

Some larger examples :