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Honaker prime  and  Sophie Germain prime

First numbers that belong to both sets:

131, 1049, 2141, 2273, 5711, 5741, 6563, 14159, 15173, 15803, 17159, 18041, 24971, 25229, 26189, 32009, 37049, 40763, 40823, 45119, 51203, 52553, 52733, 53309, 53591, 54941, 55733, 60251, 61703, 65843, 70079, 70379, 75503, 75941, 77711, 77813, 85133, 87071, 88661, 90173, 91583, 92861, 94151, 94229, 96443, 97241, 99041, 100559, 102359, 103619, 111053, 112163, 117503, 122471, 126653, 127733, 129221, 135731, 139409, 139703, 142403, 145121, 145721, 154211, 156371, 157211, 161159, 162119, 162293, 162593, 164621, 168803, 170711, 175229, 175853, 183119, 183461, 183503, 187721, 196139, 198641, 202049, 203873, 204461, 205253, 213263, 214913, 215573, 216263, 217229, 219911, 220469, 228251, 228311, 228341, 230309, 231443, 237179, 242603, 244313, 244901, 247421, 247913, 248141, 249233, 251501, 251831, 261329, 273233, 275711, 279431, 282833, 290243, 291701, 294341, 295433, 299213, 302273, 304049, 305093, 305219, 305771, 306473, 306503, 309539, 312941, 317741, 329321, 329711, 330329, 330359, 330719, 332399, 332873, 333941, 334289, 334973, 338231, 339323, 340079, 340919, 345881, 348563, 352403, 357611, 374903, 382163, 385193, 402053, 417509, 418031, 418751, 423191, 423281, 423749, 425363, 425471, 427733, 431153, 432023, 432251, 433781, 435731, 438701, 439613, 439811, 441461, 450293, 454931, 455321, 455531, 466853, 468323, 473741, 479861, 482231, 491171, 500459, 501173, 502643, 503621, 505283, 515153, 515741, 516053, 516209, 517553, 521243, 533261, 534203, 536801, 537041, 538931, 541529, 547061, 551363, 551861, 552239, 552731, 556043, 558209, 560159, 565391, 569711, 570659, 581921, 584261, 596741, 598463, 600893, 601631, 605393, 607583, 618311, 624521, 626051, 634421, 636731, 644051, 644153, 651221, 665051, 670223, 672323, 673223, 679229, 695021, 704771, 711629, 717581, 718871, 723623, 724583, 729719, 735209, 737843, 737921, 739331, 740189, 740693, 750413, 751631, 754343, 773273, 781523, 783119, 792023, 795023, 795101, 804521, 804581, 805061, 805853, 808343, 810443, 810839, 813443, 815063, 815273, 821819, 822743, 822803, 826541, 827063, 830279, 833309, 834149, 835391, 836861, 838133, 840839, 842111, 842519, 845441, 846341, 854039, 876131, 894611, 904019, 904283.

Some larger examples :