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alternating n.  and  Duffinian n.

First numbers that belong to both sets:

4, 8, 9, 16, 21, 25, 27, 32, 36, 49, 50, 63, 65, 81, 85, 98, 121, 125, 129, 143, 161, 169, 183, 185, 187, 189, 256, 301, 305, 309, 323, 325, 327, 329, 341, 343, 361, 363, 365, 381, 385, 387, 505, 525, 527, 529, 543, 549, 565, 567, 583, 589, 676, 707, 721, 723, 729, 741, 749, 763, 767, 781, 785, 901, 903, 905, 921, 923, 925, 927, 943, 949, 961, 965, 981, 985, 989, 1058, 1250, 1296, 2101, 2105, 2107, 2123, 2125, 2127, 2163, 2165, 2169, 2181, 2183, 2187, 2189, 2303, 2305, 2307, 2321, 2323, 2327, 2329, 2345, 2361, 2363, 2369, 2387, 2501, 2507, 2509, 2523, 2525, 2527, 2547, 2561, 2563, 2567, 2569, 2581, 2585, 2587, 2705, 2709, 2721, 2723, 2743, 2747, 2761, 2763, 2781, 2783, 2785, 2901, 2921, 2923, 2929, 2941, 2943, 2947, 2965, 2981, 2983, 2987, 2989, 3698, 3872, 4105, 4121, 4123, 4141, 4143, 4147, 4161, 4163, 4165, 4167, 4169, 4181, 4183, 4187, 4189, 4301, 4303, 4307, 4309, 4321, 4325, 4341, 4343, 4361, 4367, 4369, 4383, 4385, 4387, 4503, 4505, 4525, 4529, 4541, 4543, 4565, 4585, 4589, 4701, 4705, 4707, 4709, 4727, 4741, 4745, 4747, 4761, 4763, 4765, 4769, 4781, 4901, 4907, 4923, 4925, 4927, 4941, 4945, 4949, 4961, 4963, 4981, 4985, 4989, 5476, 5618, 6103, 6107, 6109, 6123, 6125, 6127, 6161, 6169, 6181, 6183, 6185, 6187, 6305, 6307, 6325, 6347, 6349, 6381, 6383, 6385, 6503, 6505, 6507, 6509, 6527, 6541, 6543, 6545, 6561, 6565, 6583, 6587, 6589, 6707, 6721, 6727, 6743, 6749, 6767, 6769, 6787, 6789, 6901, 6905, 6921, 6923, 6925, 6929, 6941, 6943, 6985, 6989, 8105, 8107, 8121, 8125, 8127, 8129, 8141, 8143, 8149, 8163, 8165, 8185, 8189, 8301, 8303, 8305, 8309, 8321.

Some larger examples :