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alternating n.  and  apocalyptic n.

First numbers that belong to both sets:

218, 278, 361, 434, 478, 529, 541, 583, 610, 612, 614, 650, 723, 747, 749, 787, 836, 838, 850, 894, 898, 925, 927, 929, 981, 983, 985, 1018, 1056, 1058, 1090, 1092, 1212, 1230, 1232, 1270, 1272, 1290, 1298, 1416, 1418, 1432, 1434, 1438, 1452, 1470, 1472, 1478, 1490, 1492, 1494, 1630, 1634, 1636, 1638, 1652, 1656, 1658, 1670, 1674, 1690, 1692, 1812, 1870, 1896, 2101, 2103, 2105, 2123, 2125, 2143, 2161, 2163, 2185, 2187, 2189, 2301, 2303, 2321, 2323, 2325, 2327, 2329, 2343, 2345, 2347, 2349, 2361, 2363, 2365, 2383, 2387, 2389, 2501, 2507, 2527, 2543, 2545, 2549, 2561, 2563, 2569, 2581, 2589, 2701, 2703, 2705, 2709, 2721, 2723, 2725, 2741, 2743, 2765, 2767, 2769, 2789, 2901, 2909, 2921, 2927, 2943, 2945, 2947, 2949, 2967, 2981, 2985, 2987, 3010, 3012, 3014, 3016, 3032, 3038, 3054, 3056, 3072, 3076, 3078, 3094, 3096, 3210, 3230, 3232, 3236, 3238, 3250, 3254, 3256, 3258, 3270, 3272, 3274, 3276, 3278, 3294, 3296, 3298, 3412, 3414, 3416, 3430, 3432, 3434, 3452, 3454, 3470, 3478, 3490, 3492, 3494, 3610, 3612, 3614, 3616, 3630, 3650, 3652, 3654, 3656, 3672, 3676, 3678, 3690, 3692, 3696, 3698, 3814, 3816, 3830, 3832, 3838, 3854, 3856, 3858, 3872, 3874, 3876, 3878, 3890, 3892, 3894, 3896, 3898, 4101, 4103, 4105, 4107, 4109, 4121, 4123, 4125, 4143, 4145, 4147, 4149, 4165, 4167, 4181, 4185, 4187, 4189, 4301, 4303, 4321, 4323, 4325, 4327, 4329, 4341, 4343, 4345, 4361, 4363, 4365, 4367, 4381, 4383, 4385, 4501, 4503, 4505, 4507, 4509, 4523, 4541, 4543, 4545, 4547, 4549, 4563, 4567, 4581, 4583, 4585, 4587, 4589, 4701, 4705, 4707, 4709, 4725, 4727, 4729, 4741, 4743, 4745, 4747, 4761, 4765, 4769, 4781, 4783, 4785, 4787, 4789, 4901, 4903, 4905.

Some larger examples :