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alternating n.  and  Proth n.

First numbers that belong to both sets:

3, 5, 9, 25, 41, 49, 65, 81, 129, 145, 161, 321, 385, 545, 705, 769, 929, 961, 2305, 2369, 2561, 2945, 6145, 6529, 6785, 8321, 8705, 8961, 10369, 12161, 12545, 12929, 14721, 16129, 16385, 18945, 30721, 32769, 34305, 34561, 38145, 50945, 54529, 54785, 56321, 58369, 70145, 72705, 92161, 94721, 96769, 98305, 210945, 214529, 232961, 236545, 250369, 252929, 258561, 272385, 278529, 296961, 414721, 416769, 432129, 450561, 456705, 476161, 492545, 610305, 630785, 636929, 656385, 670721, 672769, 696321, 698369, 816129, 834561, 850945, 876545, 892929, 1014785, 1030145, 1054721, 1056769, 1210369, 1218561, 1234945, 1296385, 1456129, 1474561, 1490945, 1634305, 1654785, 1814529, 1832961, 1890305, 3010561, 3016705, 3078145, 3094529, 3256321, 3258369, 3272705, 3436545, 3452929, 3614721, 3616769, 3676161, 3692545, 3856385, 3870721, 3872769, 5050369, 5058561, 5074945, 5218305, 5238785, 5296129, 5672961, 5816321, 5832705, 5894145, 7036929, 7098369, 7430145, 7454721, 7610369, 7618561, 7634945, 7696385, 7856129, 9052161, 9232385, 9256961, 9416705, 9478145, 9494529, 9658369, 23216129, 23650305, 23814145, 23830529, 23838721, 25018369, 25272321, 25436161, 25452545, 25616385, 25632769, 25698305, 25878529, 27074561, 27090945, 27254785, 41058305, 41238529, 41410561, 41656321, 41672705, 41836545, 41852929, 43270145, 43294721, 43450369, 43458561, 43474945, 43638785, 43696129, 45072385, 45096961, 45498369, 47816705, 49250305, 49414145, 49430529, 49438721, 49618945, 61276161, 61292545, 61456385, 61472769, 61890561, 63094785, 65634305, 65814529, 67010561, 67256321, 67272705, 67436545, 67452929, 67616769, 69058561, 69074945, 69238785, 69894145, 83230721, 83410945, 83656705, 83836929, 85032961, 85278721, 85458945, 85852161, 87490561, 87670785, 101416961, 103874561, 103890945, 109232129, 109854721, 121094145, 121274369, 121896961, 125616129, 127074305, 127238145, 127254529, 127418369, 129056769, 141410305, 143032321, 143212545, 143458305, 143638529, 145014785, 145096705, 145276929, 145670145, 145850369, 147292161, 147472385, 147898369, 161054721, 161218561, 161234945, 161890305, 163676161, 163692545, 163856385, 163872769, 165494785, 169410561, 169656321, 169672705, 169836545, 169852929, 181698561, 181878785, 185630721, 185810945, 187056129, 187432961, 187678721, 187858945, 189890561, 301072385, 301498369, 305430529, 307036161, 307232769, 307298305, 309690369, 321814529, 323256321, 323452929, 323616769, 325058561, 329416705, 341278721, 343670785, 347832321, 349470721, 349634561, 349896705, 365232129, 365854721, 367034369, 367656961.

Some larger examples :