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cyclic n.  and  balanced n.

First numbers that belong to both sets:

2, 11, 15, 19, 35, 37, 41, 139, 141, 149, 163, 177, 197, 209, 527, 535, 541, 551, 557, 563, 565, 569, 583, 587, 589, 595, 601, 611, 613, 617, 647, 653, 659, 661, 665, 677, 681, 707, 709, 713, 721, 779, 781, 787, 789, 793, 803, 805, 809, 817, 835, 865, 899, 901, 913, 929, 2095, 2103, 2141, 2157, 2165, 2191, 2203, 2215, 2219, 2221, 2227, 2229, 2251, 2261, 2281, 2319, 2327, 2333, 2343, 2347, 2357, 2381, 2387, 2389, 2393, 2405, 2417, 2443, 2453, 2467, 2473, 2481, 2501, 2505, 2513, 2587, 2589, 2599, 2603, 2611, 2617, 2643, 2659, 2661, 2699, 2701, 2707, 2713, 2723, 2729, 2737, 2755, 2761, 2785, 2823, 2827, 2829, 2837, 2841, 2851, 2857, 2885, 2897, 2913, 2949, 2953, 2977, 3009, 3095, 3101, 3115, 3147, 3149, 3157, 3161, 3173, 3221, 3235, 3269, 3273, 3281, 3335, 3341, 3347, 3349, 3353, 3365, 3377, 3395, 3397, 3401, 3409, 3461, 3473, 3489, 3521, 3595, 3605, 3619, 3621, 3651, 3653, 3657, 3665, 3715, 3729, 3745, 3777, 3845, 3849, 6695, 6719, 6721, 6767, 6791, 6793, 6799, 6857, 6881, 6883, 6911, 6947, 6959, 6983, 6989, 6991, 7001, 7013, 7017, 7027, 7031, 7039, 7091, 7097, 7099, 7115, 7117, 7123, 7145, 7151, 7153, 7163, 7179, 7187, 7193, 7197, 7201, 7223, 7231, 7241, 7243, 7247, 7251, 7257, 7273, 7343, 7361, 7367, 7369, 7397, 7433, 7445, 7469, 7477, 7487, 7499, 7517, 7559, 7607, 7703, 7715, 7751, 7793, 7799, 7801, 7811, 7823, 7827, 7835, 7837, 7841, 7849, 7877, 7913, 7945, 7949, 7957, 7967, 7971, 7993, 8063, 8069, 8071, 8087, 8089, 8095, 8117, 8123, 8135, 8147, 8159, 8161, 8173, 8197, 8203, 8213, 8219, 8223, 8227, 8239, 8245, 8255, 8279, 8285, 8287, 8297, 8309, 8311, 8315, 8317, 8321, 8327, 8331, 8335.

Some larger examples :