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alternating n.  and  cyclic n.

First numbers that belong to both sets:

1, 2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 65, 67, 69, 83, 85, 87, 89, 101, 103, 107, 109, 123, 127, 141, 143, 145, 149, 161, 163, 167, 181, 185, 187, 303, 307, 321, 323, 329, 341, 345, 347, 349, 365, 367, 383, 389, 501, 503, 509, 521, 523, 527, 541, 545, 547, 561, 563, 565, 569, 581, 583, 587, 589, 701, 703, 705, 707, 709, 721, 727, 743, 745, 749, 761, 763, 767, 769, 781, 785, 787, 789, 901, 907, 923, 929, 941, 943, 947, 949, 965, 967, 983, 985, 989, 2101, 2103, 2123, 2129, 2141, 2143, 2147, 2149, 2161, 2165, 2167, 2183, 2185, 2309, 2321, 2323, 2327, 2341, 2343, 2345, 2347, 2363, 2369, 2381, 2383, 2387, 2389, 2501, 2503, 2505, 2507, 2509, 2521, 2543, 2545, 2549, 2561, 2563, 2567, 2569, 2581, 2587, 2589, 2701, 2703, 2707, 2723, 2729, 2741, 2743, 2747, 2749, 2761, 2765, 2767, 2785, 2787, 2789, 2903, 2905, 2909, 2921, 2923, 2927, 2929, 2941, 2949, 2963, 2965, 2969, 2981, 2983, 2987, 4101, 4103, 4109, 4121, 4123, 4127, 4129, 4141, 4145, 4147, 4163, 4169, 4181, 4183, 4187, 4189, 4303, 4307, 4309, 4321, 4323, 4327, 4343, 4349, 4363, 4369, 4381, 4385, 4387, 4501, 4505, 4507, 4521, 4523, 4529, 4541, 4543, 4547, 4549, 4561, 4567, 4569, 4583, 4589, 4703, 4709, 4721, 4723, 4727, 4729, 4741, 4745, 4747, 4749, 4763, 4765, 4769, 4781, 4783, 4787, 4789, 4903, 4909, 4921, 4927, 4943, 4945, 4963, 4967, 4969, 4981, 4985, 4987, 6101, 6103, 6107, 6109, 6121, 6127, 6141, 6143, 6145, 6149, 6161, 6163, 6167, 6169, 6185, 6187, 6189, 6301, 6303, 6305, 6307, 6323, 6329, 6341, 6343, 6347, 6349, 6361, 6365, 6367, 6383, 6385, 6387, 6389, 6501, 6503, 6509, 6521, 6523.

Some larger examples :