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cyclic n.  and  weak prime

First numbers that belong to both sets:

3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647, 661, 677, 683, 691, 709, 743, 761, 773, 797, 811, 823, 829, 839, 859, 863, 883, 887, 911, 919, 941, 953, 971, 983, 997, 1013, 1021, 1033, 1039, 1051, 1063, 1069, 1093, 1097, 1109, 1129, 1153, 1171, 1193, 1201, 1217, 1231, 1237, 1259, 1279, 1283, 1291, 1303, 1307, 1321, 1327, 1373, 1381, 1409, 1429, 1433, 1439, 1453, 1459, 1483, 1489, 1493, 1499, 1531, 1553, 1559, 1571, 1583, 1601, 1609, 1613, 1621, 1627, 1637, 1669, 1699, 1709, 1723, 1759, 1789, 1811, 1831, 1873, 1879, 1889, 1913, 1933, 1951, 1979, 1999, 2003, 2017, 2029, 2039, 2069, 2083, 2089, 2099, 2113, 2131, 2143, 2161, 2179, 2207, 2213, 2221, 2239, 2243, 2251, 2269, 2273, 2297, 2311, 2341, 2351, 2357, 2383, 2393, 2399, 2423, 2441, 2447, 2477, 2543, 2551, 2557, 2593, 2621, 2633, 2659, 2663, 2689, 2693, 2699, 2713, 2719, 2731, 2753, 2777, 2791, 2803, 2837, 2843, 2861, 2887, 2909, 2917, 2927, 2939, 2957, 2971, 3001, 3023, 3041, 3049, 3067, 3083, 3089, 3121, 3137, 3169, 3191, 3209, 3221, 3229, 3253, 3259, 3271, 3301, 3323, 3331, 3347, 3361, 3373, 3391, 3413, 3463, 3469, 3499, 3517, 3529, 3533, 3541, 3547, 3559, 3583, 3593, 3617, 3623, 3643, 3673, 3677, 3701, 3709, 3739, 3769, 3779, 3797, 3803, 3823, 3833, 3853, 3863, 3881, 3889, 3911, 3919, 3923, 3931, 3947, 3967, 4003, 4007, 4021, 4027, 4051, 4057, 4079, 4093, 4099, 4111, 4129, 4133, 4139, 4159, 4177.

Some larger examples :