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cyclic n.  and  self n.

First numbers that belong to both sets:

1, 3, 5, 7, 31, 53, 97, 143, 187, 209, 211, 233, 255, 277, 299, 323, 345, 367, 389, 411, 413, 435, 457, 479, 501, 547, 569, 591, 613, 659, 681, 703, 727, 749, 771, 793, 815, 817, 839, 883, 929, 951, 973, 995, 1021, 1043, 1087, 1109, 1111, 1133, 1177, 1199, 1223, 1245, 1267, 1289, 1313, 1335, 1357, 1401, 1447, 1469, 1513, 1537, 1559, 1603, 1627, 1649, 1671, 1693, 1717, 1739, 1761, 1783, 1829, 1851, 1873, 1895, 1919, 1941, 1963, 1985, 2033, 2055, 2077, 2099, 2123, 2167, 2213, 2235, 2257, 2279, 2347, 2369, 2391, 2413, 2437, 2459, 2481, 2503, 2549, 2571, 2593, 2615, 2617, 2661, 2683, 2729, 2773, 2795, 2819, 2841, 2863, 2885, 2931, 2953, 3023, 3067, 3089, 3113, 3157, 3247, 3269, 3291, 3313, 3337, 3359, 3427, 3449, 3493, 3515, 3517, 3539, 3561, 3583, 3605, 3651, 3673, 3695, 3719, 3763, 3785, 3831, 3853, 3919, 3921, 3943, 4009, 4035, 4057, 4079, 4101, 4147, 4169, 4213, 4237, 4259, 4281, 4303, 4327, 4349, 4393, 4415, 4439, 4461, 4483, 4505, 4529, 4573, 4595, 4619, 4663, 4685, 4819, 4821, 4843, 4865, 4909, 4933, 4999, 5069, 5091, 5113, 5137, 5227, 5249, 5293, 5315, 5317, 5339, 5361, 5383, 5429, 5473, 5495, 5519, 5541, 5563, 5585, 5631, 5653, 5721, 5743, 5765, 5809, 5833, 5899, 5921, 5923, 5989, 6011, 6037, 6059, 6081, 6103, 6127, 6149, 6193, 6217, 6239, 6261, 6283, 6305, 6329, 6373, 6395, 6463, 6485, 6553, 6619, 6621, 6643, 6709, 6733, 6755, 6799, 6821, 6823, 6911, 6935, 6979, 7001, 7027, 7049, 7071, 7093, 7115, 7117, 7183, 7229, 7251, 7273, 7295, 7319, 7341, 7363, 7431, 7453, 7519, 7543, 7565, 7609, 7633, 7699, 7721, 7723, 7745, 7789, 7811, 7835, 7879, 7901, 7969, 7991, 8039, 8061, 8083, 8129, 8173, 8219.

Some larger examples :