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alternating n.  and  hoax n.

First numbers that belong to both sets:

58, 85, 94, 234, 250, 274, 361, 438, 454, 456, 476, 634, 650, 654, 690, 985, 1612, 1650, 1678, 1812, 1858, 1876, 1894, 2325, 2541, 2565, 2785, 2965, 3036, 3474, 3476, 3618, 3630, 3694, 3836, 3850, 3892, 4189, 4369, 4765, 4905, 4981, 5052, 5094, 5098, 5298, 5412, 5458, 5612, 5638, 5652, 5674, 5818, 5854, 5874, 6187, 6385, 6567, 6583, 6585, 6741, 6981, 7038, 7052, 7076, 7078, 7212, 7236, 7438, 7650, 7674, 7834, 7850, 8145, 8149, 8185, 8325, 8347, 8545, 9052, 9094, 9096, 9274, 9294, 9412, 9436, 9630, 9634, 9650, 9652, 10705, 10761, 14725, 16105, 16123, 16125, 16947, 18985, 21478, 21498, 21694, 21852, 21874, 23098, 23238, 23474, 23496, 23650, 23658, 23694, 23818, 23850, 23854, 23870, 25078, 25092, 25294, 25298, 25434, 25618, 25834, 25850, 27036, 27236, 27238, 27418, 27814, 27892, 29038, 29074, 29078, 29092, 29254, 29434, 29612, 29634, 29852, 30505, 30523, 32989, 34725, 34987, 38525, 38589, 38763, 38785, 38965, 41052, 41056, 41058, 41098, 41216, 41250, 41278, 41452, 41494, 41678, 43276, 43294, 43416, 43474, 43674, 43676, 43854, 43876, 45036, 45078, 45216, 45236, 45238, 45270, 45274, 45296, 45418, 45436, 45454, 45634, 45650, 45814, 45836, 47036, 47074, 47092, 47218, 47254, 47256, 47454, 47852, 47856, 49018, 49054, 49232, 49254, 49276, 49416, 49436, 50143, 50305, 50523, 50967, 52789, 54369, 56325, 56587, 58365, 58941, 58945, 61078, 61212, 61216, 61250, 61256, 61490, 61612, 61618, 61636, 61676, 61850, 61876, 63090, 63094, 63232, 63454, 63476, 63496, 63634, 63674, 63814, 63834, 63870, 65034, 65072, 65074, 65218, 65252, 65256, 65434, 65454, 65492, 65618, 65636, 67034, 67096, 67234, 67252, 67452, 67636, 67850, 69050, 69052, 69076, 69214, 69270, 69654, 72589, 74565, 74569, 74749, 74781, 74785, 74787, 74983, 76185, 76323, 76549, 76747, 78349, 78725, 78745, 81058, 81234, 81274, 81418, 81650, 81676, 83038, 83050, 83076, 83094, 83218.

Some larger examples :