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balanced prime  and  modest n.

First numbers that belong to both sets:

211, 733, 23333, 29333, 40111, 134333, 202999, 260999, 289999, 290033, 311111, 391999, 566999, 706999, 853999, 930101, 991999, 1000099, 1000231, 1090303, 1160303, 1423333, 1531111, 1700813, 1723333, 2011111, 2120303, 2210303, 2371111, 2400271, 3200123, 3233333, 3439999, 3830909, 3911111, 4340909, 5131111, 5693333, 5819999, 5920909, 6820909, 7000033, 7100909, 7441111, 7553333, 7583333, 7621111, 8009999, 10008547, 10010101, 10259999, 10351111, 10793333, 11423333, 12529999, 13013333, 15700813, 16012987, 16313333, 16589999, 17023333, 18713333, 20663333, 21900271, 23002849, 24053333, 24911111, 25142857, 25729999, 26473333, 27463333, 27533333, 28139999, 29179999, 31649999, 31700369, 32273333, 34009009, 35500813, 37000297, 38059999, 40015873, 40076923, 40900813, 42102439, 42109999, 42211111, 46000429, 51289999, 51619999, 52000693, 55309999, 56389999, 57329999, 59069999, 60209999, 62090909, 63419999, 70319999, 70489999, 74802439, 75109999, 77909999, 79000231, 80329999, 81029999, 82006993, 83006993, 83369999, 84511111, 86000407, 86008547, 88012987, 89002439, 89309999, 90799999, 92179999, 94909999, 95001443, 98669999, 99959999, 100333333, 101000407, 101006993, 102000259, 102233333, 103004329, 103030303, 106111111, 107037037, 107111111, 113015873, 131030303, 137000777, 140013947, 140047619, 142099999, 149902439, 155000189, 155015873, 169211111, 169499999, 170000273, 173001287, 175499999, 187133333, 193006993, 199000481, 205008547, 206711111, 210833333, 211047619, 212000777, 228733333, 233202439, 236902439, 244000429, 251009009, 252000407, 254015873, 269633333, 272001287, 272002079, 281411111, 283499999, 284002457, 290000693, 298711111, 301000481, 301411111, 302001001, 306833333, 313033333, 314002079, 316111111, 316411111, 335005291, 335010101, 342511111, 343012987, 346999999, 364511111, 365001443, 367002331, 367633333, 368411111, 370027027, 371010989, 371599999, 373003861, 375733333, 382399999, 382511111, 382633333, 386000819, 387000407, 388027027, 395333333, 398111111, 400499999, 424633333, 430012987, 430076923, 433090909, 437142857, 446005291, 460001221, 473010101, 477833333, 479142857, 480037037, 488076923, 500000909, 501369863, 502333333, 504011111, 507599999, 510811111, 511099999, 519233333, 522005291, 524001287, 529711111, 531833333, 543199999, 545002849, 547015873, 550002457, 550090909, 557142857, 562076923, 566001287, 569012987, 593333333, 604000693, 604003663, 610009009, 628004329, 640003003, 640499999, 641399999, 655076923, 661008547, 662333333, 662599999, 666533333, 673001221, 688511111, 691001287, 692015873, 692090909, 702233333, 712001221, 713333333, 722001221, 724001221, 724047619, 725699999, 726799999, 752933333, 758003003, 772027027, 772142857, 773047619, 774433333, 775733333, 776002849, 779003003, 779999999, 783002849, 785833333, 788037037, 792199999, 799999999, 801511111, 802076923, 821003663, 823001287, 827001221, 833012987.

Some larger examples :