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alternating n.  and  Smith n.

First numbers that belong to both sets:

4, 27, 58, 85, 94, 121, 274, 438, 454, 634, 636, 654, 690, 729, 852, 985, 1678, 1858, 1872, 1894, 2583, 2745, 2785, 2965, 3258, 3294, 3690, 3694, 3852, 4185, 4189, 4369, 4743, 4765, 4981, 5098, 5298, 5458, 5638, 5674, 5818, 5854, 5874, 6187, 6385, 6567, 6583, 6585, 6981, 7078, 7438, 7674, 7834, 8149, 8185, 8307, 8347, 8545, 8901, 9036, 9094, 9274, 9276, 9294, 9296, 9414, 9634, 10705, 10761, 14985, 16105, 16123, 16983, 18765, 18963, 18985, 21078, 21478, 21498, 21694, 21874, 21896, 23098, 23658, 23694, 23818, 23854, 23870, 25078, 25236, 25256, 25294, 25298, 25614, 25618, 25834, 27238, 27270, 27296, 27418, 27696, 27814, 29038, 29074, 29078, 29254, 29434, 29456, 29634, 29890, 30505, 30523, 32989, 34987, 38385, 38589, 38785, 38925, 38965, 41098, 41278, 41436, 41494, 41678, 41892, 43294, 43474, 43674, 43854, 45054, 45078, 45238, 45274, 45418, 45454, 45634, 45656, 45814, 45872, 47074, 47076, 47218, 47254, 47454, 47456, 47656, 47816, 49018, 49054, 49254, 49616, 49818, 50143, 50305, 50523, 50787, 52785, 52789, 52947, 54567, 54963, 56587, 58305, 58945, 61078, 61490, 61618, 61838, 61856, 63094, 63276, 63454, 63634, 63656, 63674, 63814, 63834, 63870, 65074, 65096, 65218, 65434, 65454, 65618, 65652, 65816, 67234, 67270, 67812, 67832, 69214, 69216, 69270, 69272, 69472, 69654, 69836, 69856, 70929, 72589, 74169, 74569, 74749, 74785, 74787, 74943, 74983, 76323, 76549, 76563, 76747, 78349, 78745, 81058, 81274, 81418, 81636, 83038, 83094, 83218, 83254, 83258, 83634, 83636, 83694, 83856, 85018, 85472, 85632, 85692, 85890, 87032, 87034, 87036, 87090, 87096, 87214, 87230, 87636, 87856, 89014, 89412, 89414, 90363, 90981, 92749, 92763, 92787, 92965, 92985, 94185, 94567, 94585, 94785, 94909, 96169, 96363, 96583, 96727, 96781, 96921, 98109, 98301, 98343, 98545, 98901, 101856, 101892, 105632, 107432, 107698, 107878, 109434, 109452, 121812, 121836, 123096, 123876, 123898, 125676.

Some larger examples :