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Proth n.  and  weak prime

First numbers that belong to both sets:

3, 13, 113, 193, 241, 353, 449, 577, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 6529, 7937, 10369, 10753, 12289, 13313, 13697, 14593, 15233, 15361, 22273, 23041, 23297, 26113, 26881, 31489, 36353, 39937, 40193, 45569, 46337, 49921, 50177, 57601, 58369, 64513, 76289, 79873, 81409, 84481, 86017, 87041, 95233, 96769, 102913, 112129, 113153, 120833, 133121, 133633, 136193, 138241, 147457, 151553, 153089, 158209, 159233, 161281, 176129, 183809, 187393, 188417, 202753, 216577, 219649, 220673, 235009, 238081, 240641, 242689, 251393, 254977, 279553, 286721, 301057, 307201, 319489, 329729, 365569, 380929, 384001, 463873, 525313, 534529, 575489, 577537, 617473, 638977, 649217, 667649, 673793, 675841, 694273, 695297, 706561, 721921, 724993, 737281, 765953, 772097, 778241, 783361, 786433, 790529, 795649, 808961, 814081, 817153, 858113, 875521, 925697, 946177, 959489, 964609, 986113, 995329, 1032193, 1038337, 1062913, 1067009, 1073153, 1075201, 1079297, 1097729, 1103873, 1124353, 1146881, 1179649, 1196033, 1198081, 1239041, 1251329, 1288193, 1318913, 1355777, 1370113, 1376257, 1437697, 1454081, 1492993, 1509377, 1517569, 1533953, 1595393, 1628161, 1720321, 1724417, 1732609, 1751041, 1785857, 1841153, 1865729, 2101249, 2119681, 2123777, 2148353, 2209793, 2236417, 2314241, 2377729, 2414593, 2455553, 2482177, 2486273, 2516993, 2553857, 2572289, 2574337, 2639873, 2652161, 2660353, 2756609, 2768897, 2875393, 2910209, 2936833, 2979841, 3022849, 3053569, 3072001, 3076097, 3082241, 3115009, 3133441, 3194881, 3198977, 3219457, 3348481, 3360769, 3383297, 3414017, 3420161, 3438593, 3452929, 3489793, 3526657, 3543041, 3598337, 3612673, 3635201, 3647489, 3684353, 3721217, 3735553, 3850241, 3870721, 3893249, 3919873, 3944449, 3960833, 3985409, 3993601, 4104193, 4120577, 4165633, 4206593, 4263937, 4300801, 4423681, 4427777, 4435969, 4452353, 4550657, 4620289, 4866049, 4882433, 5050369, 5189633, 5234689, 5324801, 5337089, 5345281, 5410817, 5492737, 5517313, 5591041, 5656577, 5763073, 5779457, 5824513, 5849089, 5865473, 5951489, 6008833, 6111233, 6144001, 6156289, 6254593, 6307841, 6381569, 6393857, 6402049, 6537217, 6574081, 6578177, 6725633, 6737921, 6750209, 6782977, 6885377, 6942721, 6946817, 7057409, 7139329, 7168001, 7200769, 7340033, 7360513, 7413761, 7434241, 7475201, 7630849, 7643137, 7659521.

Some larger examples :