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balanced prime  and  de Polignac n.

First numbers that belong to both sets:

373, 977, 3637, 4013, 4691, 5303, 6073, 7583, 7841, 9473, 10253, 10607, 11411, 15161, 16787, 17047, 17327, 19483, 19949, 20201, 21163, 23767, 23893, 24097, 24413, 26723, 28597, 29269, 29873, 30313, 30449, 30643, 32993, 34171, 34631, 36677, 40933, 41017, 41947, 42533, 44543, 45433, 45757, 46147, 48023, 48533, 51461, 52529, 54673, 56773, 57119, 57853, 58067, 62213, 71843, 74471, 74897, 77951, 79867, 80923, 81439, 81689, 81749, 83437, 83443, 86981, 89107, 89113, 89983, 91297, 91303, 91387, 93053, 93083, 94427, 96431, 103567, 104021, 104113, 104953, 105607, 105613, 109897, 110681, 111521, 112181, 112939, 113921, 117757, 119033, 119563, 120097, 120823, 121327, 121553, 125441, 125737, 126211, 126317, 127663, 132911, 134213, 134639, 135089, 137393, 138107, 139079, 140351, 140527, 140983, 142019, 144247, 144253, 144611, 145121, 145589, 146197, 148633, 150203, 150571, 151687, 160117, 163567, 167759, 170081, 171329, 172433, 172993, 174749, 179167, 179369, 179743, 182549, 182893, 184517, 185267, 186253, 187373, 188437, 189473, 191027, 192991, 196291, 197927, 202387, 202859, 203227, 204013, 206203, 209743, 213827, 216731, 219613, 223253, 223481, 229739, 230227, 237619, 237821, 240491, 241327, 245753, 246787, 246913, 247421, 247901, 249329, 250919, 252163, 252431, 253537, 255127, 256307, 261089, 263083, 264637, 266977, 267569, 270967, 271357, 279613, 282487, 282697, 285457, 285463, 288577, 291089, 295259, 295277, 295837, 297257, 297397, 297439, 302983, 303187, 305131, 311957, 318259, 318431, 320107, 325993, 327667, 329281, 330749, 331147, 332699, 334183, 335261, 338573, 341623, 346349, 346553, 347759, 348553, 349477, 353099, 353641, 354307, 357823, 358993, 362081, 364961, 366133, 370399, 370477, 372859, 379283, 381817, 389369, 393103, 393611, 393677, 394987, 395909, 396091, 396937, 397951, 398441, 399577, 400429, 401689, 402043, 402071, 403043, 406661, 406981, 408637, 409027, 414697, 415147, 423103, 427717, 430499, 432037, 432097, 434933, 435541, 437953, 439799, 443347, 445877, 446603, 446951, 451337, 456439, 456523, 457241, 458651, 463297, 466567, 467497, 468647, 470501, 471521, 472133, 473911, 476513, 480761, 481807, 484079.

Some larger examples :