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cyclic n.  and  interprime n.

First numbers that belong to both sets:

15, 69, 217, 393, 473, 501, 515, 667, 705, 771, 803, 879, 885, 933, 1003, 1167, 1207, 1243, 1313, 1465, 1517, 1527, 1537, 1551, 1563, 1695, 1795, 1817, 1895, 1941, 2001, 2157, 2191, 2229, 2391, 2405, 2453, 2463, 2627, 2661, 2685, 2703, 2859, 2913, 2933, 2955, 3039, 3055, 3073, 3265, 3337, 3353, 3453, 3495, 3531, 3651, 3665, 3849, 3921, 3937, 3945, 3957, 3995, 4085, 4119, 4189, 4353, 4403, 4415, 4521, 4535, 4685, 4697, 4755, 4771, 4883, 5001, 5045, 5249, 5267, 5631, 5727, 5739, 5781, 5833, 5933, 6009, 6107, 6117, 6157, 6185, 6207, 6333, 6409, 6613, 6645, 6667, 6735, 6749, 6891, 6969, 6999, 7033, 7091, 7115, 7233, 7341, 7597, 7689, 7747, 7805, 7835, 7847, 7913, 7957, 8097, 8135, 8185, 8341, 8437, 8445, 8507, 8517, 8679, 8691, 8781, 8805, 8813, 8877, 8985, 9035, 9079, 9097, 9167, 9193, 9215, 9233, 9249, 9267, 9527, 9569, 9607, 9637, 9669, 9727, 9741, 9797, 9823, 9831, 9865, 9877, 9885, 10085, 10155, 10187, 10217, 10245, 10263, 10281, 10363, 10461, 10507, 10599, 10645, 10717, 10785, 10897, 11015, 11077, 11103, 11267, 11283, 11293, 11305, 11477, 11523, 11595, 11627, 11851, 11947, 12039, 12085, 12131, 12219, 12233, 12273, 12295, 12385, 12405, 12417, 12533, 12639, 12665, 12847, 12871, 12963, 13021, 13161, 13235, 13263, 13303, 13353, 13389, 13405, 13493, 13641, 13703, 14151, 14395, 14631, 14655, 14663, 14691, 14817, 14829, 14837, 14943, 14963, 15067, 15087, 15155, 15261, 15441, 15613, 15635, 15735, 15755, 15841, 16045, 16059, 16133, 16251, 16449, 16465, 16557, 16613, 16735, 16753, 16799, 16817, 16837, 16953, 16971, 17065, 17197, 17319, 17437, 17503, 17529, 17695, 17733, 17755, 17817, 17845, 17857, 17897, 17965, 18001, 18159, 18255, 18263, 18295, 18309, 18335, 18347, 18407, 18471, 18487, 18519, 18547, 18649, 18707, 18737, 18905, 18933, 18953, 19243, 19295, 19389, 19397.

Some larger examples :