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modest n.  and  twin primes

First numbers that belong to both sets:

13, 19, 29, 59, 103, 109, 199, 311, 433, 599, 809, 811, 1033, 1699, 1933, 1999, 2027, 2111, 2999, 3299, 4799, 5009, 5099, 6199, 6299, 7333, 8009, 8599, 8999, 10009, 10037, 10303, 10333, 13999, 17027, 22037, 22111, 23027, 25999, 26111, 28111, 30011, 49333, 55333, 58111, 60101, 61333, 68111, 70999, 79999, 80909, 85333, 86111, 94111, 101111, 101999, 131111, 160033, 164999, 170099, 170101, 179999, 184999, 230999, 247999, 268999, 272999, 280099, 293999, 301333, 301999, 313333, 322999, 326999, 328333, 331999, 379999, 389999, 410999, 419999, 430909, 443999, 448999, 450101, 496999, 500909, 511111, 512999, 524999, 536999, 580303, 611111, 616999, 620909, 652999, 659999, 662999, 670099, 691111, 694999, 710909, 721111, 733333, 769999, 773999, 781999, 788999, 794999, 800999, 818999, 820909, 850303, 868999, 880303, 880909, 890999, 904999, 907999, 910099, 920999, 932999, 970999, 973333, 980909, 989999, 1000037, 1000039, 1000429, 1003003, 1003367, 1100041, 1171111, 1273333, 1363333, 1381111, 1393333, 1420303, 1480909, 1483333, 1500041, 1600813, 1603333, 1609999, 1693333, 1841111, 2000351, 2002331, 2003861, 2025641, 2030909, 2051111, 2200813, 2202439, 2390909, 2480909, 2623333, 2710909, 2810909, 2833333, 3001001, 3005291, 3043333, 3149999, 3181111, 3271111, 3350909, 3402439, 3481111, 3740909, 3800369, 4000037, 4000039, 4000693, 4100909, 4121111, 4139999, 4211111, 4231111, 4270909, 4441111, 4500271, 4520909, 4580909, 4679999, 4720909, 4900123, 4940909, 5000111, 5041111, 5110909, 5201111, 5381111, 5502439, 5539999, 5551111, 5570909, 5800369, 6000481, 6011111, 6013333, 6080909, 6089999, 6223333, 6260909, 6400369, 6490909, 6500369, 6569999, 6613333, 6650909, 6671111, 6689999, 6730909, 6751111, 6760909, 6781111, 6811111, 6851111, 6941111, 6959999, 7031111, 7040909, 7047619, 7271111, 7279999, 7351111, 7459999, 7519999, 7550909, 7589999, 7639999, 7730909, 7751111, 7861111, 7900813, 7990909, 8000099, 8000117, 8009009, 8071111, 8170909, 8291111, 8300909, 8351111, 8609999, 8659999, 8680909, 8731111, 8819999, 8849999, 8863333, 8959999, 8981111, 9061111, 9239999, 9371111, 9451111, 9829999, 9929999, 9943333, 10001443, 10002439, 10003003, 10153333, 10219999, 10300271, 10339999, 10513333, 10543333, 10579999, 10709999, 11000111, 11069999, 11089999, 11593333, 11602439, 11623333, 11629999, 11719999, 11869999.

Some larger examples :