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nialpdrome  and  Sophie Germain prime

First numbers that belong to both sets:

2, 3, 5, 11, 41, 53, 83, 431, 443, 641, 653, 743, 761, 911, 953, 4211, 5333, 5441, 6521, 6551, 7211, 7433, 7541, 7643, 8111, 8663, 8741, 9221, 42221, 44111, 54443, 65111, 66431, 76421, 77543, 77711, 84431, 84443, 86111, 86441, 87221, 88661, 88811, 94421, 96443, 97511, 98321, 98621, 98711, 99551, 99761, 444443, 533321, 553211, 555521, 644411, 653111, 662111, 664421, 665111, 743111, 764111, 764321, 766541, 766553, 775553, 777431, 777761, 842111, 855521, 883331, 884111, 885551, 886433, 887333, 888533, 888773, 953321, 954221, 955211, 955541, 963311, 965411, 973331, 975521, 985433, 986411, 986543, 987533, 988511, 988661, 995531, 996311, 997511, 997541, 997553, 998111, 998633, 998831, 999221, 999521, 999611, 999653, 3221111, 3222221, 4211111, 4433333, 5444321, 5555321, 6433211, 6553331, 6555443, 6643211, 7553333, 7632221, 7666553, 7733321, 7754321, 7765211, 7774433, 7775543, 8554421, 8555441, 8654333, 8764331, 8764433, 8831111, 8865443, 8875211, 8876531, 8876663, 8877653, 8886551, 8886653, 8887661, 8888531, 8888651, 9543311, 9554333, 9644111, 9666551, 9772211, 9774221, 9777653, 9855311, 9855533, 9872111, 9888881, 9976541, 9977321, 9977651, 9987311, 9987443, 9988211, 9997661, 9998633, 9999653, 33332111, 43332221, 43333211, 44444321, 53333321, 54222221, 54322211, 54443111, 55433333, 55532111, 64333331, 64433321, 65552111, 65554211, 65555411, 66432221, 66444221, 66533321, 66542111, 66555311, 66644111, 66654221, 66666311, 66666653, 75333311, 75544433, 76322111, 76444211, 76554431, 76555211, 76655333, 76655531, 77433221, 77522111, 77632211, 77655551, 77754221, 77766641, 77777321, 83333111, 84431111, 85333331, 86533211, 86543321, 87311111, 87322211, 87554333, 87555521, 87666311, 87755543, 87764321, 87774311, 87777551, 88422221, 88554311, 88754411, 88773221, 88863311, 88864211, 88866611, 88885421, 88888643, 88888841, 95522111, 95554421, 96666653, 97433333, 97633331, 97654421, 97762211, 98422211, 98441111, 98554331, 98654411, 98733221, 98754431, 98765531, 98777621, 98854421, 98855111, 98872211, 98875541, 98883311, 99543221, 99554111, 99666653, 99732221, 99761111, 99765521, 99766631, 99774431, 99775211, 99777533, 99855533, 99873311, 99885521, 99886331, 99887633, 99888221, 99954443, 99976421, 99983333, 99984221, 99999551, 99999611, 433322111, 444433331, 444444443, 554111111, 555332111, 654444221, 655543211, 655544111, 663222221, 665542211, 666333221, 666444431, 666533321, 666533333, 666644411, 733222211, 744443321, 754222211.

Some larger examples :