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modest n.  and  Sophie Germain prime

First numbers that belong to both sets:

23, 29, 89, 233, 509, 809, 911, 1499, 1733, 2003, 2399, 2699, 3299, 5333, 5399, 6899, 8111, 17333, 26111, 35111, 35999, 38333, 44111, 65111, 68111, 71999, 86111, 90011, 101333, 101999, 137999, 149333, 164999, 188999, 200033, 209333, 221999, 240101, 260999, 266999, 272333, 287333, 293999, 305999, 329999, 335999, 410999, 419999, 470303, 470999, 524999, 533999, 599999, 602999, 620099, 620303, 620909, 650099, 704999, 720101, 731999, 770909, 788999, 830099, 854999, 890303, 932999, 956999, 974999, 980909, 1271111, 1283333, 1343333, 1400303, 1553333, 1841111, 1859999, 2000039, 2003861, 2051111, 2069999, 2210303, 2261111, 2489999, 2639999, 2651111, 2720303, 2723333, 2780909, 2921111, 3221111, 3410909, 3479999, 3593333, 3629999, 3911111, 4100909, 4211111, 4223333, 4283333, 4433333, 4583333, 4700123, 4730909, 4763333, 4901111, 4940909, 5000111, 5001443, 5002331, 5231111, 5300909, 5369999, 5633333, 5900123, 6002849, 6011111, 6083333, 6113333, 6502439, 6650909, 6689999, 6710909, 6920909, 7400369, 7553333, 7583333, 7589999, 7619999, 7853333, 7943333, 8000231, 8009999, 8189999, 8201111, 8483333, 8579999, 8789999, 8831111, 8879999, 8960909, 9001001, 9869999, 9893333, 10000271, 10463333, 10469999, 10571111, 10829999, 11000189, 11001443, 11030303, 11300813, 11602439, 12089999, 12449999, 13013333, 13043333, 13229999, 13289999, 13403333, 13433333, 13469999, 13600271, 13709999, 13793333, 14273333, 14423333, 15002849, 15029999, 15149999, 15179999, 15263333, 15500813, 15539999, 15713333, 15983333, 16349999, 16373333, 16379999, 16439999, 16463333, 16589999, 16673333, 16829999, 17000999, 17003333, 17010989, 17033333, 17303333, 17429999, 17600813, 17633333, 17759999, 17813333, 17909999, 17969999, 18713333, 18983333, 19100369, 19229999, 19403333, 20000693, 20393333, 20543333, 20600369, 21500813, 21593333, 21749999, 21773333, 22049999, 22289999, 22439999, 22613333, 23303333, 23903333, 23963333, 24023333, 24389999, 24569999, 24800813, 25379999, 25613333, 25679999, 25739999, 26000099, 26000231, 26001221, 26010989, 26099999, 26600813, 26902439, 27209999, 27419999, 27479999, 27502439, 27533333, 28523333, 28739999, 29002439, 29249999, 29669999, 29900369, 29903333, 29999999, 30005291, 30202439, 30863333, 31199999, 31253333, 32369999, 32411111, 32879999, 33323333, 34733333, 35602439, 35819999, 36311111, 36719999, 37199999, 38429999, 38711111, 38902439, 39200813, 39233333, 40289999, 40529999, 41000819, 41003663, 41009009, 41159999, 41549999, 41849999, 42059999, 42269999, 42389999, 42419999, 43109999, 44600813, 45011111.

Some larger examples :