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congruent n.  and  Proth n.

First numbers that belong to both sets:

5, 13, 41, 65, 145, 161, 257, 353, 609, 1025, 1217, 2113, 2817, 2945, 3265, 3713, 3841, 4033, 4481, 6529, 6657, 8705, 9089, 9345, 9601, 9729, 11265, 12161, 12545, 12673, 12929, 13441, 14209, 14593, 15105, 15361, 15617, 16385, 22785, 24577, 25345, 25601, 27393, 28161, 29185, 30721, 30977, 32513, 33025, 33537, 35329, 35841, 36865, 39169, 40961, 44289, 44545, 44801, 45313, 46081, 48641, 49153, 50177, 50689, 52993, 53761, 54529, 55297, 55809, 58881, 60417, 61953, 62465, 63745, 64001, 64769, 67073, 75777, 82945, 85505, 90625, 97281, 110593, 116225, 117761, 118785, 119297, 120321, 121857, 129025, 129537, 130561, 136705, 137217, 146433, 157697, 160769, 162305, 168961, 178689, 189441, 193025, 201729, 202753, 204289, 210945, 217601, 224257, 227329, 231425, 238081, 239105, 239617, 240129, 240641, 241665, 250881, 253953, 254465, 259585, 260097, 262145, 273409, 277505, 288769, 294913, 296961, 323585, 326657, 331777, 335873, 349185, 364545, 382977, 414721, 415745, 420865, 462849, 482305, 489473, 495617, 522241, 560129, 590849, 602113, 617473, 625665, 626689, 633857, 640001, 641025, 648193, 651265, 665601, 672769, 676865, 686081, 696321, 714753, 751617, 755713, 762881, 763905, 770049, 775169, 787457, 790529, 794625, 805889, 818177, 820225, 825345, 826369, 828417, 840705, 868353, 877569, 881665, 883713, 889857, 894977, 909313, 932865, 943105, 961537, 973825, 986113, 1002497, 1012737, 1014785, 1032193, 1042433, 1054721, 1062913, 1112065, 1116161, 1148929, 1173505, 1214465, 1267713, 1320961, 1327105, 1349633, 1353729, 1359873, 1388545, 1398785, 1417217, 1464321, 1466369, 1470465, 1474561, 1501185, 1533953, 1536001, 1548289, 1554433, 1566721, 1585153, 1587201, 1599489, 1626113, 1640449, 1646593, 1665025, 1710081, 1777665, 1787905, 1806337, 1818625, 1835009, 1839105, 1859585, 1865729, 1916929, 1947649, 1953793, 2025473, 2039809, 2111489, 2117633, 2131969, 2179073, 2238465, 2275329, 2279425, 2304001, 2310145, 2312193, 2340865, 2355201, 2371585, 2428929, 2502657, 2521089, 2541569, 2639873, 2684929, 2721793, 2777089, 2811905, 2830337, 2838529, 2842625, 2844673, 2863105, 2897921, 2899969, 2918401, 2926593, 2934785, 2951169, 2957313, 2963457, 2981889, 2985985, 3104769, 3117057, 3176449, 3186689.

Some larger examples :