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Proth n.  and  twin primes

First numbers that belong to both sets:

3, 5, 13, 17, 41, 193, 241, 641, 1153, 2113, 2689, 3329, 4481, 9857, 11777, 12161, 14081, 14593, 15361, 17921, 22273, 23041, 25601, 26113, 26881, 49409, 49921, 58369, 65537, 95233, 102913, 114689, 119297, 133633, 138241, 168449, 202753, 219649, 226817, 232961, 235009, 238081, 249857, 279553, 326657, 365569, 455681, 495617, 498689, 514049, 557057, 572417, 617473, 675841, 694273, 724993, 737281, 783361, 786433, 795649, 817153, 836609, 875521, 928769, 995329, 1017857, 1032193, 1054721, 1062913, 1124353, 1370113, 1374209, 1417217, 1517569, 1589249, 1662977, 1751041, 2101249, 2369537, 2387969, 2414593, 2660353, 2664449, 2707457, 2775041, 2787329, 2936833, 3022849, 3133441, 3168257, 3194881, 3348481, 3360769, 3612673, 3690497, 3819521, 3870721, 3942401, 4046849, 4165633, 4169729, 4304897, 4464641, 4476929, 4771841, 4866049, 4907009, 5234689, 5398529, 5517313, 5591041, 5763073, 5767169, 5849089, 6144001, 6320129, 6574081, 6873089, 7438337, 7667713, 7729153, 7745537, 7790593, 7806977, 7925761, 8114177, 8183809, 8286209, 8466433, 9023489, 9035777, 9080833, 9166849, 9441281, 9502721, 9646081, 9711617, 10067969, 10076161, 10186753, 10559489, 10571777, 10989569, 11046913, 11304961, 11358209, 12390401, 12451841, 12902401, 13037569, 13152257, 13283329, 13344769, 13496321, 13631489, 13877249, 14241793, 14307329, 14499841, 15052801, 15228929, 15523841, 16269313, 16355329, 16384001, 16650241, 16699393, 16957441, 17252353, 18006017, 18358273, 18497537, 18964481, 19701761, 20512769, 21200897, 22118401, 22241281, 22437889, 22601729, 23961601, 24035329, 24199169, 24764417, 26361857, 26419201, 26484737, 27353089, 27475969, 28499969, 28557313, 28827649, 29286401, 30179329, 30515201, 30785537, 32440321, 32612353, 32686081, 33538049, 33832961, 34357249, 34570241, 34643969, 34652161, 35135489, 35430401, 35684353, 36175873, 36544513, 37060609, 37232641, 37273601, 37601281, 37896193, 38133761, 38256641, 38526977, 38993921, 39493633, 40370177, 41730049, 42582017, 43294721, 43745281, 43859969, 44105729, 44285953, 44802049, 44851201, 45137921, 45506561, 45588481, 45662209, 46497793, 46522369, 47382529, 48906241, 49274881, 49438721, 49635329, 49709057, 49938433, 51191809, 51650561, 51658753, 52224001, 52412417, 52461569, 53501953, 53567489, 54747137, 56770561, 57573377, 57917441, 58531841, 58613761, 59277313, 59424769, 59523073, 60252161, 61480961, 61636609, 62537729, 63594497, 63700993, 63815681, 64438273, 64749569, 64946177, 65241089, 65290241, 65781761, 66347009, 66969601, 67649537, 68190209, 68567041, 70483969, 71811073.

Some larger examples :