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cyclic n.  and  Proth n.

First numbers that belong to both sets:

3, 5, 13, 17, 33, 41, 65, 97, 113, 145, 161, 177, 193, 209, 241, 257, 321, 353, 449, 481, 545, 577, 641, 673, 705, 769, 865, 929, 1153, 1217, 1345, 1409, 1473, 1537, 1601, 1729, 1793, 1921, 1985, 2049, 2113, 2177, 2369, 2497, 2561, 2689, 2753, 2881, 3009, 3073, 3137, 3265, 3329, 3457, 3521, 3585, 3649, 3713, 3777, 3841, 4033, 4097, 4353, 4481, 4865, 4993, 5249, 5377, 5633, 5761, 6017, 6145, 6401, 6529, 6785, 6913, 7169, 7297, 7553, 7681, 7937, 8065, 8321, 9089, 9217, 9473, 9601, 9857, 9985, 10113, 10369, 10753, 11009, 11393, 11521, 11777, 12161, 12289, 12417, 12545, 12673, 12801, 12929, 13057, 13313, 13441, 13569, 13697, 14081, 14593, 14849, 14977, 15233, 15361, 15617, 15745, 16001, 16385, 16897, 17153, 17665, 17921, 18433, 18689, 19457, 19969, 20481, 20737, 20993, 21761, 22017, 22273, 22529, 23041, 23297, 23809, 24065, 24577, 24833, 25089, 25345, 25601, 26113, 26881, 27137, 27649, 28417, 28673, 29185, 29441, 29953, 30209, 30721, 30977, 31233, 31489, 31745, 32001, 32257, 32513, 33281, 33793, 34049, 34561, 34817, 36097, 36353, 36609, 37121, 37633, 37889, 38145, 38401, 38657, 39937, 40193, 40961, 41473, 41729, 42241, 42497, 43009, 43265, 43777, 44033, 44801, 45057, 45313, 45569, 46337, 46849, 47617, 48385, 48641, 49153, 49409, 49921, 50177, 50433, 50689, 50945, 51713, 51969, 52481, 52737, 52993, 53249, 53761, 54017, 54529, 54785, 55553, 56065, 56321, 56833, 57089, 57601, 57857, 58369, 59137, 59393, 60161, 60673, 60929, 61185, 61441, 61697, 62209, 62977, 63233, 64001, 64257, 64513, 64769, 65281, 65537, 67073, 68609, 70145, 70657, 71681, 72193, 73217, 73729, 74753, 75265, 76289, 76801, 79361, 79873, 80385, 80897, 81409, 82433, 82945, 83969, 84481, 86017, 87041, 87553, 88577, 89601, 90113, 91137, 91649, 92161, 93185, 93697, 94721, 95233, 96257, 96769, 97793, 98817, 99329, 99841, 100353, 100865, 101377, 102401.

Some larger examples :