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nialpdrome  and  strong prime

First numbers that belong to both sets:

11, 41, 71, 97, 311, 331, 431, 521, 541, 631, 641, 751, 821, 853, 877, 881, 991, 2111, 4211, 4421, 4441, 5431, 5441, 6211, 6311, 6421, 6521, 6551, 6653, 7211, 7321, 7331, 7411, 7741, 7753, 8111, 8311, 8443, 8521, 8641, 8663, 8731, 8831, 8861, 8887, 9221, 9311, 9431, 9511, 9533, 9643, 9733, 9811, 9851, 9883, 9941, 33311, 42221, 44111, 54311, 55331, 55541, 63311, 63331, 64433, 65111, 65551, 66221, 66643, 72211, 72221, 74311, 74411, 74441, 75431, 75521, 75553, 76333, 76421, 76541, 76631, 77521, 77543, 77611, 77641, 77711, 77743, 77761, 77773, 84211, 84221, 84421, 84431, 85331, 85411, 86111, 86311, 86441, 86531, 87211, 87221, 87421, 87553, 87641, 88211, 88321, 88411, 88643, 88651, 88661, 88721, 88741, 88771, 88811, 88843, 88853, 88861, 94321, 94421, 95311, 95441, 96211, 96221, 96443, 96553, 96643, 96661, 97711, 97771, 98221, 98321, 98443, 98533, 98621, 98663, 98711, 98773, 98887, 99431, 99551, 99643, 99661, 99761, 99871, 99877, 99961, 333331, 422111, 443221, 444443, 533111, 542111, 543311, 555521, 611111, 632221, 642211, 644431, 644443, 653111, 653311, 654221, 654421, 655211, 655331, 655531, 655541, 664421, 665111, 666431, 666511, 666643, 733321, 733331, 742111, 742211, 743111, 743321, 743333, 744431, 754333, 755551, 762211, 763111, 764111, 764431, 765521, 765533, 766211, 766421, 766531, 766541, 766553, 775441, 775553, 776551, 777431, 777541, 777641, 777661, 777731, 777761, 822221, 832211, 842111, 843331, 844111, 844421, 853211, 855221, 855511, 855521, 864221, 865211, 865321, 866431, 874331, 875443, 876431, 876643, 877333, 877531, 877651, 877771, 884111, 885421, 885551, 886421, 886511, 886643, 886663, 887311, 887441, 887533, 887651, 887743, 888443, 888533, 888631, 888653, 888721, 888751, 888773, 922211, 952111, 953221, 953321, 953333, 954221, 954433, 955211, 955333, 955433, 963211, 964333, 965443, 965551, 966221, 966431, 966521, 966653, 972221, 973321, 973331, 974411, 974431, 975421, 975521, 975551, 976553, 977411, 982211, 984421, 985331, 985433, 986333, 986411, 986533, 986633, 987433.

Some larger examples :