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esthetic n.  and  unprimeable n.

First numbers that belong to both sets:

898, 3232, 3234, 5454, 5456, 5676, 5678, 7654, 7656, 7898, 9898, 21232, 21234, 23210, 23212, 23232, 23234, 34345, 34565, 43210, 43212, 43432, 43434, 45454, 45456, 65454, 65456, 67876, 67878, 76765, 78765, 87654, 87656, 87898, 89876, 89878, 98765, 101010, 101012, 101210, 101212, 101232, 101234, 121210, 121212, 121232, 121234, 212345, 232345, 234565, 321210, 321212, 323210, 323212, 323432, 323434, 323454, 323456, 343454, 343456, 434545, 454345, 454565, 456545, 456565, 543432, 543434, 543454, 543456, 545454, 545456, 545676, 545678, 565432, 565434, 565676, 565678, 654565, 656545, 656765, 676545, 676565, 765454, 765456, 767654, 767656, 767876, 767878, 767898, 787654, 787656, 787676, 787678, 787898, 789876, 789878, 789898, 876545, 878765, 987676, 987678, 987876, 987878, 987898, 989898, 1012345, 1232345, 1234565, 2121210, 2121212, 2123432, 2123434, 2123454, 2123456, 2343210, 2343212, 2343432, 2343434, 2343454, 2343456, 2345432, 2345434, 2345676, 2345678, 3232345, 3234345, 3234545, 3234565, 3434345, 3434545, 3434565, 3454565, 4323454, 4323456, 4343232, 4343234, 4343432, 4343434, 4345432, 4345434, 4345454, 4345456, 4345676, 4345678, 4543210, 4543212, 4543432, 4543434, 4543454, 4543456, 4545654, 4545656, 4545676, 4545678, 4565432, 4565434, 4565454, 4565456, 4567654, 4567656, 4567676, 4567678, 5432345, 5434545, 5434565, 5456545, 5654345, 5656565, 5656765, 5676545, 6543210, 6543212, 6543432, 6543434, 6545454, 6545456, 6545676, 6545678, 6565454, 6565456, 6565654, 6565656, 6565676, 6565678, 6567676, 6567678, 6765454, 6765456, 6765654, 6765656, 6767676, 6767678, 6767876, 6767878, 6787654, 6787656, 6787676, 6787678, 6787898, 6789876, 6789878, 6789898, 7656545, 7656565, 7676545, 7676765, 7678765, 7876545, 8765454, 8765456, 8765676, 8765678, 8767654, 8767656, 8767898, 8787654, 8787656, 8787876, 8787878, 8789876, 8789878, 8987676, 8987678, 8987876, 8987878, 8987898, 8989898, 9876545, 9878765.