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esthetic n.  and  sphenic n.

First numbers that belong to both sets:

78, 345, 434, 654, 678, 987, 1010, 2121, 2343, 2345, 3434, 3454, 4323, 4345, 4543, 4565, 5654, 5678, 6565, 6567, 6789, 7654, 7678, 7898, 8787, 8789, 8987, 9878, 12121, 12345, 21234, 32123, 34323, 34565, 54321, 54345, 54565, 56545, 56787, 65454, 65654, 67898, 76765, 78767, 78987, 87878, 87898, 89898, 98987, 121010, 123454, 212343, 212345, 232345, 234321, 234543, 234545, 234565, 323210, 323234, 343454, 345454, 432321, 434545, 454321, 454323, 454345, 454565, 456565, 456789, 543454, 545654, 545678, 565454, 565678, 567678, 567878, 654545, 654567, 656543, 656545, 656765, 676565, 676787, 765434, 765654, 767654, 767878, 787678, 787898, 789878, 789898, 876565, 876765, 898765, 987898, 1010101, 1012121, 1012345, 1210101, 1212101, 1212343, 1232121, 1232321, 1234345, 2101010, 2321234, 2323454, 2343234, 2343434, 2343454, 2345678, 3212321, 3212345, 3232101, 3232345, 3234323, 3234567, 3434323, 3434345, 3434543, 3434545, 3434567, 3454545, 3454565, 3454567, 4321210, 4323434, 4323454, 4343234, 4543234, 4543454, 4545434, 4565654, 4567898, 5432323, 5434323, 5434545, 5454565, 5456543, 5456545, 5456765, 5654345, 5654545, 5656767, 5656787, 5676789, 5678765, 6545434, 6545454, 6545654, 6565654, 6567898, 6765434, 6765454, 6767678, 6767898, 6787654, 6789878, 6789898, 7656567, 7656765, 7676543, 7676545, 7676767, 7676789, 7678765, 7678987, 7876565, 7876765, 7876767, 7878787, 8765434, 8765654, 8765678, 8767654, 8789878, 8987654, 8987898, 8989898, 9876543, 9876567, 9878989, 9898765, 9898987, 9898989, 10123210, 10123234, 10123454, 12121210, 12121234, 12123434, 12343234, 12345654, 21010121, 21012121, 21012123, 21012343, 21212321, 21232343, 21234323, 21234545, 21234565, 23212343, 23234323, 23234345, 23432123, 23432321, 23432323, 23432345, 23454345, 23454545, 23456767, 23456787, 32101010, 32101210, 32121010, 32321234, 32323210, 32323234, 32345678, 34321010, 34321234, 34345678, 34543434, 34545454, 34565654, 43210101, 43212101, 43212121, 43212323, 43232123, 43232321, 43232345, 43234543, 43234545, 43432121, 43432123, 43434543, 43434565, 43454543, 43454545, 43456543, 43456567, 43456767, 45432121, 45432321, 45434323, 45434343, 45454323, 45456545, 45456789, 45654321, 45654545, 45656543, 45676543, 45676545, 45676565, 45678787, 54323210, 54323454, 54343210, 54543234, 54545434, 54565678, 56565434, 56765454, 56765678, 56789878, 65432121, 65432321, 65434323, 65434343, 65434567, 65454321, 65454567, 65456565, 65456789, 65654321, 65654323.

Some larger examples :