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cyclic n.  and  Ulam n.

First numbers that belong to both sets:

1, 2, 3, 11, 13, 47, 53, 69, 77, 87, 97, 131, 145, 177, 197, 209, 221, 241, 319, 339, 341, 409, 431, 451, 485, 607, 673, 685, 695, 739, 751, 781, 893, 949, 983, 991, 1037, 1079, 1103, 1157, 1167, 1169, 1257, 1313, 1335, 1387, 1433, 1465, 1489, 1509, 1531, 1553, 1643, 1707, 1709, 1721, 1765, 1853, 1919, 1941, 1985, 2093, 2095, 2117, 2249, 2327, 2371, 2393, 2447, 2481, 2513, 2581, 2589, 2633, 2723, 2787, 2789, 2831, 2833, 2897, 2899, 2921, 2987, 3029, 3031, 3041, 3043, 3065, 3085, 3107, 3109, 3131, 3217, 3239, 3263, 3371, 3373, 3395, 3415, 3473, 3495, 3527, 3547, 3593, 3605, 3649, 3669, 3671, 3691, 3991, 4057, 4101, 4121, 4153, 4187, 4211, 4233, 4297, 4343, 4363, 4409, 4451, 4453, 4517, 4519, 4531, 4619, 4729, 4883, 4903, 4927, 4969, 5035, 5057, 5059, 5081, 5269, 5291, 5357, 5489, 5531, 5533, 5597, 5599, 5663, 5765, 5797, 5873, 6029, 6049, 6107, 6239, 6283, 6391, 6437, 6457, 6459, 6479, 6481, 6523, 6569, 6635, 6721, 6735, 6833, 6865, 6911, 6931, 7043, 7195, 7197, 7217, 7219, 7297, 7427, 7459, 7471, 7473, 7537, 7559, 7583, 7603, 7691, 7727, 7735, 7737, 7781, 7979, 8011, 8057, 8077, 8099, 8101, 8143, 8167, 8255, 8453, 8509, 8531, 8539, 8573, 8617, 8639, 8749, 8773, 8817, 8873, 8917, 8947, 8969, 8971, 8983, 9013, 9071, 9137, 9179, 9193, 9311, 9313, 9357, 9377, 9379, 9401, 9445, 9509, 9511, 9553, 9577, 9619, 9643, 9663, 9665, 9721, 9743, 9809, 9851, 9853, 9917, 9919, 9941, 9961, 9983, 9985, 10085, 10247, 10249, 10271, 10349, 10369, 10391, 10393, 10415, 10457, 10459, 10501, 10523, 10537, 10547, 10567, 10657, 10679, 10691, 10757, 10779, 10831, 10909, 10931, 10977, 10997, 11087, 11099, 11107, 11195, 11197, 11219, 11273, 11603, 11627, 11647, 11735.

Some larger examples :