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balanced prime  and  Sophie Germain prime

First numbers that belong to both sets:

5, 53, 173, 593, 653, 1103, 1223, 1511, 2903, 2963, 4409, 5303, 6263, 6323, 7823, 7841, 9473, 10253, 12653, 13463, 15161, 15803, 18341, 18731, 22259, 25673, 27773, 29873, 30449, 34631, 36821, 38183, 40283, 41603, 44543, 54443, 54773, 58979, 59393, 60083, 60509, 62213, 66431, 67169, 68699, 71843, 73523, 78173, 82493, 82793, 84443, 85049, 85121, 92003, 92363, 93053, 101483, 110459, 111509, 111773, 112181, 112583, 120563, 122099, 122273, 126839, 127733, 128153, 128483, 131933, 133571, 135893, 137393, 137849, 139703, 140351, 145121, 163853, 165443, 167729, 167759, 169283, 171329, 174143, 174749, 182261, 193373, 194723, 196739, 198491, 202841, 202859, 206033, 206483, 212099, 216113, 219071, 220919, 225983, 229739, 243203, 247421, 247451, 252341, 253481, 258539, 260999, 269063, 276833, 289853, 291293, 295943, 297623, 302573, 303293, 304943, 306011, 311009, 311183, 317783, 318431, 318641, 320693, 325883, 335579, 338573, 343823, 346751, 350453, 353813, 354323, 356579, 358829, 359279, 361643, 364103, 368129, 371561, 390353, 396479, 403043, 410513, 411563, 412019, 418751, 422063, 434243, 439811, 443123, 443153, 446951, 449381, 450293, 450893, 458063, 471521, 472883, 473513, 476243, 482393, 484079, 487463, 488723, 488909, 491423, 491633, 493133, 497963, 504209, 509429, 510299, 513899, 517073, 526139, 531353, 531983, 532391, 535511, 536621, 540173, 540383, 552053, 558599, 558881, 560459, 569759, 576683, 581891, 583013, 583403, 605009, 605921, 608831, 611939, 613829, 620519, 622613, 630023, 644123, 644513, 655643, 657269, 657911, 662339, 668273, 678773, 682751, 687773, 693683, 694409, 702269, 704183, 705713, 706913, 713183, 725519, 732509, 733283, 734543, 734813, 735473, 748361, 750713, 750803, 752639, 758963, 759131, 759953, 766763, 776201, 781619, 783743, 784103, 788549, 802793, 803189, 804113, 813401, 814031, 816239, 817913, 819653, 825479, 825509, 829463, 832493, 837503, 839981, 841283, 844463, 847103, 847163, 848579, 855131, 860753, 862331, 865253, 871571, 871901, 882593, 883061, 911129, 912239, 912269, 912941, 913103, 916463, 919871, 921743, 927149, 928643, 932021, 951053, 955919, 960983, 978773, 985109, 985493, 987491, 987593.

Some larger examples :