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esthetic n.  and  prime n.

First numbers that belong to both sets:

23, 43, 67, 89, 101, 787, 4567, 12101, 12323, 12343, 32321, 32323, 34543, 54323, 56543, 56767, 76543, 78787, 78989, 210101, 212123, 234323, 234343, 432121, 432323, 432343, 434323, 454543, 456767, 654323, 654343, 678767, 678989, 876787, 878789, 878987, 878989, 898787, 898987, 1012321, 1210123, 1212121, 1234543, 3210101, 3210121, 3212101, 3212123, 3212323, 3232321, 3432101, 3454343, 3456767, 5432321, 5434321, 5434343, 5454343, 5654323, 5654567, 5656789, 5676767, 5676787, 5678767, 5678789, 5678989, 7654567, 7656787, 7676567, 7678787, 7678789, 7876567, 7876789, 7878989, 7898767, 7898789, 21212101, 21232121, 23210101, 23232101, 23234567, 23434343, 23456567, 23456789, 43212343, 43232323, 43234343, 43432321, 45432323, 45434321, 45454543, 45454567, 45456787, 45676567, 45676789, 65434543, 65456543, 65654543, 67654567, 67656767, 67678789, 87676789, 89876767, 89876789, 101012321, 101210101, 101212343, 101232121, 101232343, 121012123, 121012343, 121232101, 121232123, 121234543, 123210121, 123232121, 321012121, 321212321, 321232321, 321234343, 323210101, 323212321, 323232121, 323232323, 323234321, 323234323, 323234543, 323456767, 343212101, 343212323, 343232101, 343234321, 343432121, 343454323, 343454567, 343456543, 345432343, 345434567, 345454567, 345456767, 345656543, 345676543, 345678787, 543232121, 543432101, 543434321, 543456787, 545432101, 545432123, 545432323, 545434343, 545456567, 545656543, 545656567, 565434343, 565454543, 565654321, 565676543, 565676767, 565678787, 567676789, 567876767, 765432323, 765434543, 765454343, 765456787, 765656767, 765678787, 765678989, 767654543, 767656543, 767656567, 787676567, 787678987, 787898987, 789878767, 789898787, 987654323, 987656767, 987876767, 987878987, 987898789, 989876567, 989898989, 2101210121, 2101210123, 2101212343, 2101232123, 2101234321, 2121010123, 2121212101, 2121212321, 2121212323, 2123210123, 2123212121, 2123212123, 2123432123, 2123434343, 2123456767, 2321210123, 2321212343, 2321234323, 2323212343, 2323234567, 2323434343, 2323454321, 2323456543, 2323456787, 2343212321, 2343232321, 2343234343, 2343434321, 2343434323, 2345432123, 2345454343, 2345456567, 2345656567, 4321012123, 4321212323, 4323454321, 4343232121, 4343232343, 4343234543, 4343432123, 4343434321, 4345654543, 4345678789, 4543212101, 4543234567, 4543432321, 4543454543, 4543454567, 4543456787, 4545432343, 4545434321, 4545676567, 4545676787, 4545676789, 4565432321, 4565454343, 4565456789, 4565654567, 4565656567, 4565656787, 4565676767, 4567678789, 4567898989, 6543212101, 6543212323, 6543432121, 6543434323, 6543454543, 6543456767, 6545432323, 6545654567, 6565454321, 6565454323, 6565654321, 6567676543, 6567676567, 6567876767, 6567878987, 6567898787, 6567898789, 6765432101, 6765434321, 6765434323, 6765454543, 6765456767, 6765654323, 6765656767, 6767898989, 6787654343, 6787656787, 6787676789, 6789876767, 6789876787, 6789878987, 6789878989, 6789898987, 8765432321, 8765454323.

Some larger examples :