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nialpdrome  and  twin primes

First numbers that belong to both sets:

3, 5, 7, 11, 31, 41, 43, 61, 71, 73, 311, 421, 431, 433, 521, 641, 643, 661, 811, 821, 881, 883, 2111, 3331, 4421, 5441, 5443, 5521, 6551, 6553, 6661, 7211, 7331, 7333, 8221, 8431, 8821, 8861, 8863, 9421, 9431, 9433, 9631, 9721, 9931, 22111, 33331, 42221, 43321, 54421, 55331, 55333, 55441, 63311, 65521, 72221, 74411, 75211, 75541, 76421, 76541, 76543, 76651, 77551, 77711, 83221, 84221, 85331, 85333, 86111, 86531, 86533, 87221, 87511, 87541, 87631, 87641, 87643, 87721, 88661, 88663, 88811, 92221, 94111, 94441, 95441, 95443, 96221, 96331, 97651, 98321, 98641, 98711, 98731, 99721, 99991, 322111, 332221, 422111, 511111, 543311, 555421, 555521, 611111, 653111, 654221, 665111, 666431, 666433, 721111, 732211, 733331, 733333, 742211, 754111, 755311, 765211, 766111, 766541, 766543, 776221, 777421, 777431, 777433, 777641, 777643, 822221, 842111, 843211, 864221, 865211, 866311, 866641, 875521, 876331, 876431, 876433, 877111, 877621, 883111, 884311, 885551, 885553, 888661, 888871, 932221, 944431, 955441, 966211, 966661, 973331, 973333, 975521, 975551, 975553, 977411, 977611, 982211, 985531, 987211, 988111, 988321, 988651, 988861, 992111, 994321, 995551, 996211, 996631, 997111, 997651, 997741, 998651, 998653, 999331, 999431, 999433, 999611, 999961, 3222211, 3222221, 3333311, 4211111, 4441111, 5332211, 5422211, 5444111, 5444321, 5544221, 5551111, 5552221, 5553211, 5553311, 5554331, 5554333, 5555321, 6422111, 6643211, 6643321, 6652211, 6653221, 6655541, 6655543, 7322111, 7322221, 7422221, 7443211, 7443221, 7444441, 7533221, 7544431, 7555321, 7555421, 7644431, 7644433, 7655521, 7664311, 7665311, 7666621, 7732211, 7743221, 7744441, 7751111, 7754321, 7765211, 7766321, 7766441, 7766443, 7773211, 7776311, 8442221, 8633111, 8643211, 8663311, 8664421, 8664431, 8664433, 8665541, 8665543, 8731111, 8733311, 8755321, 8764321, 8773321, 8775331, 8775541, 8776211, 8776541, 8776543, 8777521, 8854411, 8863331, 8863333, 8866441, 8877331, 8882221, 8885311, 8885411, 8885531, 8885533, 8886331, 8887631, 8887633, 8887661, 8887663, 8888851, 9443311, 9544111, 9555311, 9632111, 9654331, 9665431, 9666311, 9743311, 9755411, 9764221, 9765211, 9774211, 9775541, 9775543.

Some larger examples :