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cyclic n.  and  lucky n.

First numbers that belong to both sets:

1, 3, 7, 13, 15, 31, 33, 37, 43, 51, 67, 69, 73, 79, 87, 115, 127, 133, 141, 151, 159, 163, 193, 211, 223, 235, 241, 259, 267, 283, 303, 307, 319, 321, 331, 339, 349, 367, 391, 393, 409, 415, 421, 427, 433, 451, 463, 487, 511, 517, 519, 535, 537, 541, 553, 559, 577, 583, 591, 601, 613, 619, 631, 643, 673, 679, 685, 699, 717, 727, 739, 745, 769, 781, 787, 805, 823, 883, 885, 895, 933, 937, 957, 991, 997, 1009, 1021, 1039, 1041, 1057, 1087, 1093, 1105, 1117, 1123, 1147, 1167, 1189, 1201, 1203, 1219, 1231, 1245, 1249, 1261, 1285, 1291, 1303, 1309, 1329, 1339, 1357, 1387, 1401, 1417, 1441, 1459, 1471, 1473, 1495, 1501, 1543, 1563, 1567, 1579, 1585, 1597, 1639, 1645, 1663, 1693, 1723, 1749, 1765, 1777, 1797, 1801, 1819, 1831, 1879, 1915, 1921, 1923, 1933, 1941, 1945, 1959, 1963, 1977, 1987, 2001, 2031, 2053, 2059, 2065, 2083, 2095, 2113, 2173, 2215, 2221, 2239, 2251, 2257, 2281, 2283, 2311, 2317, 2323, 2335, 2343, 2395, 2407, 2419, 2427, 2461, 2467, 2473, 2479, 2491, 2505, 2545, 2557, 2563, 2571, 2587, 2589, 2593, 2599, 2647, 2661, 2671, 2689, 2755, 2773, 2785, 2787, 2797, 2815, 2821, 2823, 2827, 2841, 2845, 2851, 2887, 2899, 2905, 2913, 2923, 2953, 2971, 2977, 2983, 3007, 3031, 3037, 3039, 3049, 3055, 3073, 3091, 3097, 3109, 3121, 3133, 3163, 3183, 3187, 3199, 3223, 3229, 3235, 3259, 3301, 3307, 3313, 3405, 3409, 3427, 3433, 3439, 3453, 3487, 3489, 3495, 3499, 3559, 3571, 3579, 3595, 3607, 3613, 3621, 3655, 3661, 3669, 3687, 3697, 3709, 3727, 3763, 3781, 3793, 3811, 3847, 3849, 3865, 3889, 3931, 3943, 3991, 4003, 4033, 4035, 4045, 4081, 4129, 4189, 4195.

Some larger examples :