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junction n.  and  Proth n.

First numbers that belong to both sets:

113, 513, 1217, 1409, 2113, 2305, 3009, 6017, 7425, 8321, 9217, 11521, 12417, 13313, 14209, 17921, 22017, 24321, 26113, 28929, 30209, 33025, 34817, 37121, 39937, 41217, 43009, 44033, 45825, 47617, 48129, 51713, 52225, 54017, 56833, 58625, 60929, 62721, 65025, 69633, 73729, 86529, 90625, 99329, 107521, 111617, 124417, 129025, 137217, 141313, 145921, 150017, 154113, 158721, 159233, 162817, 163329, 167937, 171009, 172033, 176129, 180225, 184833, 193025, 197633, 201217, 205825, 214017, 218625, 222721, 235521, 248321, 252417, 257025, 265217, 274433, 295937, 300033, 308225, 312321, 329729, 333825, 342017, 359425, 363521, 389121, 393217, 398337, 401409, 406529, 410625, 428033, 436225, 440321, 457729, 461825, 470017, 487425, 491521, 496641, 504833, 534529, 556033, 564225, 585729, 602113, 619521, 623617, 628737, 632833, 641025, 658433, 662529, 679937, 684033, 688129, 692225, 700417, 717825, 721921, 726017, 727041, 730113, 735233, 756737, 760833, 786433, 790529, 824321, 845825, 854017, 855041, 859137, 863233, 884737, 901121, 918529, 922625, 948225, 952321, 957441, 961537, 978945, 983041, 987137, 991233, 1003521, 1025025, 1029121, 1033217, 1054721, 1089537, 1093633, 1101825, 1148929, 1153025, 1157121, 1161217, 1187841, 1191937, 1204225, 1255425, 1314817, 1323009, 1349633, 1353729, 1409025, 1417217, 1421313, 1447937, 1452033, 1456129, 1460225, 1515521, 1554433, 1609729, 1613825, 1622017, 1648641, 1652737, 1699841, 1708033, 1716225, 1720321, 1767425, 1771521, 1814529, 1865729, 1908737, 1912833, 1921025, 1959937, 1964033, 1968129, 1972225, 1998849, 2019329, 2023425, 2074625, 2117633, 2121729, 2168833, 2172929, 2177025, 2181121, 2236417, 2240513, 2275329, 2334721, 2369537, 2373633, 2428929, 2433025, 2437121, 2441217, 2467841, 2471937, 2476033, 2480129, 2535425, 2574337, 2629633, 2633729, 2684929, 2689025, 2693121, 2701313, 2727937, 2732033, 2736129, 2740225, 2787329, 2791425, 2834433, 2885633, 2902017, 2928641, 2932737, 2979841, 2983937, 2988033, 2992129, 3004417, 3039233, 3043329, 3094529, 3102721, 3149825, 3153921, 3158017, 3162113, 3188737, 3192833, 3201025, 3205121, 3256321, 3260417, 3287041, 3295233, 3303425, 3354625, 3389441, 3393537, 3401729, 3448833, 3452929, 3457025, 3461121, 3487745, 3491841, 3500033, 3504129, 3555329, 3614721, 3649537, 3653633, 3708929, 3713025, 3717121, 3721217.

Some larger examples :