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balanced prime  and  inconsummate n.

First numbers that belong to both sets:

173, 563, 3307, 3733, 4409, 4597, 5303, 5393, 6977, 10607, 10853, 12653, 14747, 16937, 16987, 17327, 17477, 19577, 23333, 25391, 26177, 26393, 26987, 27773, 30449, 30977, 34877, 36677, 36821, 39863, 44543, 44741, 47507, 50153, 54443, 56093, 56437, 58237, 59393, 61637, 62207, 63697, 65957, 67511, 68099, 68699, 73127, 75641, 82463, 82493, 84137, 85121, 86137, 86981, 89107, 89393, 94433, 100937, 103567, 103669, 104207, 111521, 112181, 113921, 117437, 119267, 121007, 121421, 129209, 134333, 139703, 140351, 142369, 142427, 142553, 143159, 143699, 145037, 145121, 147347, 150407, 154067, 161807, 162257, 163211, 163847, 163853, 165443, 167759, 168737, 169199, 169837, 170081, 176207, 179807, 179969, 180497, 181439, 183587, 184727, 187367, 191027, 192161, 192743, 195407, 196187, 200177, 200357, 201611, 201809, 202841, 202859, 213827, 214457, 217733, 223481, 228587, 228677, 229133, 234287, 240491, 241817, 243233, 243527, 243857, 245747, 245771, 248057, 249329, 249659, 250037, 250091, 250919, 252887, 253481, 256307, 258539, 258977, 260999, 261587, 263083, 265813, 267587, 270749, 272417, 273067, 274277, 276359, 276833, 279137, 284813, 287327, 287801, 289727, 290033, 291089, 293087, 293207, 295259, 295727, 297257, 303587, 304517, 304757, 305867, 306011, 306491, 307267, 307283, 310237, 311009, 311957, 318419, 318431, 319433, 320107, 320477, 322403, 326687, 327017, 327337, 327667, 336157, 336983, 340387, 341287, 342359, 346751, 349177, 349337, 349913, 353069, 353699, 355847, 357683, 357983, 358697, 361643, 361967, 362027, 362081, 363557, 363767, 364103, 365231, 366727, 367027, 367699, 370003, 370021, 371633, 371837, 374117, 376477, 380753, 381401, 387083, 393611, 393677, 393779, 395909, 398347, 401237, 401987, 405247, 417947, 420677, 420737, 424163, 428759, 429503, 431147, 440543, 441563, 445877, 446951, 448373, 450893, 451337, 458651, 458987, 465821, 472133, 474917, 475859, 477881, 478427, 478937, 480527, 487463, 488909, 490277, 496211, 496493, 497963, 499397, 501037, 505067, 505607, 513047, 513899, 517067, 526937, 527063, 529307, 531337, 533453, 546881, 549707, 550691, 551219, 555143, 557027, 558881, 559577, 559907, 562181, 566999, 569759, 570677.

Some larger examples :