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Sophie Germain prime  and  zygodrome

First numbers that belong to both sets:

11, 44111, 77711, 88811, 1117799, 1188899, 3377999, 4433333, 4440011, 4499933, 5522999, 7774433, 7799111, 9991133, 9991199, 11177711, 11777333, 22233311, 22244333, 33344999, 44411333, 44555999, 55511999, 55577111, 66644111, 77000999, 88844999, 88866611, 110099933, 111188333, 111555533, 111779933, 111881111, 111882299, 112226633, 113300111, 114443333, 115511111, 115577333, 116663399, 116665511, 117779999, 118886699, 119977733, 119990033, 221166611, 222225599, 222244433, 222449999, 222554411, 222773399, 222779933, 222993311, 224441111, 224444333, 226655333, 226661111, 226667711, 226668833, 226668899, 227778899, 228833399, 228881111, 229992299, 330011333, 330022211, 330044999, 333337799, 333775511, 334400333, 334499999, 335522333, 336644411, 336677111, 337773311, 337777799, 338855333, 339922211, 339933311, 339944999, 441115511, 441116699, 444113333, 444113399, 444339911, 444440099, 444446633, 444779999, 444885533, 444885599, 446699933, 447779999, 553322333, 553332233, 553335599, 555566699, 558877733, 660044111, 661119911, 661177733, 662227733, 663344333, 664447733, 666222233, 666644411, 666661133, 666773333, 667788833, 669997733, 772228811, 773333399, 775544333, 775588811, 776633999, 777119933, 777221111, 777331133, 777440099, 777551111, 777991199, 777993311, 778811333, 778877999, 779966333, 881122211, 882200999, 882211199, 883355111, 884444111, 884477333, 886668833, 887722211, 887775533, 888333311, 888555533, 888779933, 888779999, 888800999, 888844433, 889955999, 990011111, 993337799, 993377111, 996644333, 997700999, 997733333, 999776699, 999944999, 999977711, 999996611, 1100000333, 1100888333, 1110001199, 1110077333, 1110077711, 1110077999, 1114400999, 1117744499, 1117766999, 1119944333, 1133377799, 1133399999, 1144411199, 1155533111, 1155544433, 1155599999, 1166000333, 1166633999, 1177700033, 1188866999, 1188899111, 2222333111, 2224441133, 2224444499, 2226667799, 2226699911, 2227700099, 2227799999, 2229977111, 2255111333, 2255777333, 2288222333, 3322244333, 3322277999, 3330044999, 3332222111, 3334441133, 3335554499, 3336644999, 3337744499, 3337770011, 3339911999, 3339944999, 3344222111, 3366644999, 3377744333, 4433355533, 4433444111, 4433555333, 4440001199, 4440088811, 4444422233, 4444433111, 4444466699, 4444499333, 4445551133, 4448880011, 4455511199, 4466777333, 4488833333, 4488855599, 4488877733, 4499333999, 4499955599, 5544422111, 5544488333, 5550044999, 5551100099, 5551122299, 5552221133, 5554411133, 5554466999, 5555333111, 5555777333, 5555777999, 5557711199, 5557799333, 5558881199, 5558886611, 5566688111, 5599988111, 6600033311, 6611444111, 6622211111, 6662229911, 6665522999, 6688844111, 7700077799, 7700111999, 7772224433, 7774446611, 7778884499, 7799555999, 8811188999, 8844455111, 8844488111, 8855111333, 8855666999, 8866622111, 8866688111, 8882288111, 8884499111, 8888222333, 8888666999, 9911133311, 9922211333, 9944222111, 9955511333, 9966611111, 9966633311, 9977744411, 9977744999, 9988899911, 9990007799, 9991177799, 9992222111, 9995550011, 9996661133, 9999988811.