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esthetic n.  and  self n.

First numbers that belong to both sets:

121, 323, 345, 1210, 1212, 1234, 2123, 3212, 21212, 34543, 34565, 54321, 54343, 54545, 54567, 87876, 87898, 89878, 234321, 234343, 234545, 234567, 323432, 323454, 343232, 343434, 343456, 432321, 432343, 434321, 434323, 434345, 454323, 543210, 543234, 898987, 989876, 989898, 1010123, 1012101, 1210101, 2123432, 2123454, 2323232, 2323434, 2323456, 2343210, 2343234, 3456543, 3456565, 3456767, 3456789, 4321232, 4323210, 4323234, 4343212, 5454543, 5454565, 5456545, 5456567, 5654343, 5654545, 5654567, 6767876, 6767898, 6787654, 6787676, 6787878, 7654321, 7654345, 8767654, 8767676, 8767878, 8787656, 8787678, 10101212, 10121010, 12101010, 12121012, 21010101, 23454543, 23454565, 23456545, 23456567, 32101012, 32345654, 32345676, 34345432, 34345454, 34345656, 34345678, 34543432, 34543454, 34545432, 34545434, 34545456, 34565434, 43434543, 43434565, 43454343, 43454545, 43454567, 45434343, 45434545, 45434567, 45454321, 45454345, 45654323, 54343432, 54343454, 54345432, 54345434, 54345456, 54543232, 54543432, 54543434, 54543456, 54545434, 56543210, 56543232, 56543234, 56543434, 65432343, 65434321, 65434345, 65454323, 67878987, 67898765, 67898787, 67898989, 76543212, 76543234, 76789876, 76789898, 78787876, 78787898, 78789878, 78987654, 78987676, 78987878, 87678987, 87878765, 87878787, 87878989, 87898767, 87898789, 89876765, 89876787, 89878767, 89878789, 98767876, 98767898, 98787654, 98787676, 98787878, 98987654, 98987656, 98987678, 101232321, 101232343, 101234321, 101234323, 101234345, 121212321, 121212343, 121232121, 121232321, 121232323, 121232345, 121234323, 123212121, 123212321, 123212323, 123212345, 123232121, 123232123, 123232323, 123432101, 123432123, 210101010, 210121012, 212101012, 212345654, 212345676, 232345432, 232345454, 232345656, 232345678, 234343432, 234343454, 234345432, 234345434, 234345456, 234543232, 234543432, 234543434, 234543456, 234545434, 321012321, 321012343, 321212121, 321212321, 321212323, 321212345, 321232121, 321232123, 321232323, 323210121, 323212121, 323212123, 323212323, 323232101, 323232123, 343210121, 343210123, 343212101, 343212123, 345456765, 345456787, 345656543, 345656565, 345656767, 345656789, 345676543, 345676545, 345676567, 432343432, 432343454, 432345432, 432345434, 432345456, 434323432, 434323454, 434343232, 434343432, 434343434, 434343456, 434345434, 434543210, 434543232, 434543234, 434543434, 454323232, 454323432, 454323434, 454323456, 454343210, 454343232, 454343234, 454343434, 454543212, 454543234, 456787876, 456787898, 456789878, 543210101, 543210123, 543456765, 543456787, 545456543, 545456565, 545456767, 545456789, 545654543, 545654565, 545656543, 545656545, 545656567, 545676545, 565454543, 565454565, 565456543, 565456545, 565456567, 565654343, 565654543, 565654545, 565654567, 565656545, 567654321.

Some larger examples :