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nialpdrome  and  weak prime

First numbers that belong to both sets:

3, 7, 31, 43, 61, 73, 83, 421, 433, 443, 643, 661, 743, 761, 773, 811, 863, 883, 887, 911, 941, 953, 971, 983, 997, 2221, 3221, 3331, 4111, 5333, 5443, 5521, 5531, 6221, 6553, 6661, 7333, 7433, 7541, 7621, 7643, 8221, 8431, 8543, 8741, 8753, 8761, 8821, 8863, 9421, 9433, 9521, 9551, 9631, 9661, 9721, 9743, 9833, 9887, 9931, 9973, 22111, 33211, 33331, 43321, 43331, 44221, 54331, 54421, 55333, 55411, 55441, 55511, 63211, 64333, 65521, 65543, 66533, 66541, 66553, 66653, 73331, 75211, 75511, 75541, 76441, 76511, 76543, 76651, 77431, 77551, 77621, 77731, 83221, 83311, 85333, 85531, 86533, 87433, 87443, 87511, 87541, 87631, 87643, 87721, 87743, 87751, 88663, 88873, 88883, 92111, 92221, 94111, 94331, 94441, 95111, 95443, 95531, 96331, 97511, 97553, 97651, 97777, 98411, 98543, 98641, 98731, 98873, 99611, 99721, 99733, 99833, 99881, 99971, 99991, 322111, 332221, 444421, 511111, 522211, 533321, 553211, 554431, 555221, 555421, 633221, 644411, 653321, 655111, 655511, 662111, 663331, 664211, 664331, 664441, 665221, 665311, 666433, 666541, 721111, 732211, 733333, 743221, 752111, 754111, 754211, 754421, 755311, 755333, 755441, 764321, 765211, 766111, 766321, 766333, 766511, 766543, 766651, 775531, 776221, 776521, 776651, 777221, 777421, 777433, 777551, 777643, 777743, 843211, 844321, 844433, 852211, 854333, 854431, 854443, 855311, 855331, 855431, 864211, 865511, 866221, 866311, 866443, 866641, 866653, 875521, 875543, 876311, 876331, 876433, 876443, 876611, 876653, 876661, 877111, 877321, 877411, 877543, 877621, 877661, 883111, 883331, 884311, 884321, 884441, 885331, 885553, 886433, 886541, 886651, 887333, 887543, 887633, 887641, 888211, 888431, 888541, 888661, 888761, 888871, 888887, 911111, 932221, 933221, 943321, 944431, 953111, 955441, 955511, 955541, 963311, 963331, 964433, 965411, 965533, 966211, 966631, 966661, 971111, 973333, 974443, 975553, 976211, 976411, 976621, 976643, 977521, 977611, 977761, 984211, 985531, 986543, 986641, 987211, 987541, 987631, 988111, 988321, 988511, 988651.

Some larger examples :