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odious n.  and  Proth n.

First numbers that belong to both sets:

13, 25, 41, 49, 81, 97, 145, 161, 193, 241, 289, 321, 385, 481, 545, 577, 641, 737, 769, 865, 929, 961, 1089, 1153, 1281, 1473, 1537, 1729, 1857, 1921, 2113, 2177, 2305, 2497, 2561, 2753, 2881, 2945, 3073, 3265, 3393, 3457, 3649, 3713, 3841, 4033, 4225, 4353, 4609, 4993, 5121, 5505, 5761, 5889, 6145, 6529, 6785, 6913, 7297, 7425, 7681, 8065, 8321, 8449, 8705, 9089, 9217, 9601, 9857, 9985, 10241, 10625, 10881, 11009, 11393, 11521, 11777, 12161, 12289, 12673, 12929, 13057, 13441, 13569, 13825, 14209, 14465, 14593, 14849, 15233, 15361, 15745, 16001, 16129, 16641, 16897, 17409, 18177, 18433, 19201, 19713, 19969, 20481, 21249, 21761, 22017, 22785, 23041, 23553, 24321, 24577, 25345, 25857, 26113, 26881, 27137, 27649, 28417, 28929, 29185, 29697, 30465, 30721, 31489, 32001, 32257, 33025, 33281, 33793, 34561, 34817, 35585, 36097, 36353, 36865, 37633, 38145, 38401, 39169, 39425, 39937, 40705, 40961, 41729, 42241, 42497, 43265, 43521, 44033, 44801, 45313, 45569, 46081, 46849, 47105, 47873, 48385, 48641, 49153, 49921, 50433, 50689, 51457, 51713, 52225, 52993, 53505, 53761, 54273, 55041, 55297, 56065, 56577, 56833, 57601, 57857, 58369, 59137, 59393, 60161, 60673, 60929, 61441, 62209, 62721, 62977, 63745, 64001, 64513, 65281, 66049, 66561, 67585, 69121, 69633, 71169, 72193, 72705, 73729, 75265, 76289, 76801, 78337, 78849, 79873, 81409, 81921, 83457, 84481, 84993, 86529, 87041, 88065, 89601, 90625, 91137, 92161, 93697, 94209, 95745, 96769, 97281, 98305, 99841, 100865, 101377, 102913, 103425, 104449, 105985, 107009, 107521, 108545, 110081, 110593, 112129, 113153, 113665, 115201, 115713, 116737, 118273, 118785, 120321, 121345, 121857, 122881, 124417, 125441, 125953, 127489, 128001, 129025, 130561, 131585, 132097, 133121, 134657, 135169, 136705, 137729, 138241, 139265, 140801, 141825, 142337, 143873, 144385, 145409, 146945, 147457, 148993, 150017, 150529, 152065, 152577, 153601, 155137, 156161, 156673.

Some larger examples :