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esthetic n.  and  gapful n.

First numbers that belong to both sets:

121, 1010, 1210, 1212, 2121, 2323, 3210, 3232, 3434, 3456, 4343, 4545, 5454, 5656, 6565, 6767, 7676, 7878, 8787, 8989, 9898, 12345, 23232, 34545, 101010, 101210, 101232, 101234, 121010, 121210, 121212, 123210, 123432, 123456, 212121, 232323, 321210, 323232, 343232, 343434, 345456, 434343, 434565, 454321, 454545, 545454, 565434, 565656, 567878, 656565, 676767, 765432, 767676, 787878, 878787, 898989, 989898, 1012121, 1012323, 1210121, 1212101, 1212321, 1212345, 1232121, 1234321, 1234545, 3432345, 3434343, 3456565, 4321212, 6543232, 9878765, 10101010, 10101012, 10101210, 10121010, 10121210, 10121232, 10123210, 10123212, 10123456, 12101010, 12101210, 12101232, 12121010, 12121210, 12121212, 12123210, 12123432, 12123454, 12123456, 12321010, 12321012, 12321210, 12323210, 12323232, 12323234, 12323456, 12343210, 12343212, 12343456, 12345432, 12345456, 12345678, 21010101, 21212121, 21234543, 23232321, 23232323, 32121210, 32123232, 32123234, 32321010, 32323232, 32343210, 32343232, 32345676, 34323210, 34323232, 34343232, 34343434, 34345656, 34543232, 34545456, 34565454, 34567878, 43434343, 43454565, 43456545, 45432123, 45434565, 45454545, 45456567, 45654345, 45656787, 54343432, 54345654, 54545454, 54545456, 54565434, 56543454, 56565656, 65432321, 65454543, 65454545, 65656565, 67676765, 67676767, 76543454, 76545432, 76565654, 76765454, 76767676, 76767678, 76787654, 78787878, 78989898, 87676565, 87878787, 89898989, 98989898, 101232345, 101234565, 121010123, 121212345, 121234545, 123212323, 123232343, 123234345, 123432345, 123434565, 123454545, 123456567, 123456765, 210101012, 210101232, 210123232, 212323232, 212343456, 232101012, 232101232, 232123232, 232321012, 232321232, 232323212, 232323432, 232343232, 232345432, 234323232, 234543232, 234545432, 234565656, 321232345, 323234345, 323434545, 323454565, 343212345, 343454545, 345432101, 345434565, 345456545, 345676765, 432323456, 432343212, 434565432, 454343232, 454567676, 456545676, 456565456, 543234567, 545454323, 567656787, 987656765, 989898765, 1010101010, 1010101210, 1010101212, 1010121010, 1010121012, 1010121210, 1010123210, 1010123232, 1010123456, 1012101010, 1012101012, 1012101210, 1012121010, 1012121210, 1012121232, 1012123210, 1012123212, 1012123434, 1012123456, 1012321010, 1012321210, 1012321212, 1012323210, 1012323432, 1012323456, 1012343210, 1012343232, 1012343234, 1012343456, 1012345456, 1210101010, 1210101012, 1210101210, 1210121010, 1210121210, 1210121232, 1210121234, 1210123210, 1210123212, 1210123456, 1212101010, 1212101210, 1212101232, 1212121010, 1212121210, 1212121212, 1212123210, 1212123234, 1212123432, 1212123456, 1212321010, 1212321012, 1212321210, 1212323210, 1212323232, 1212323434, 1212323456, 1212343210, 1212343212, 1212343454, 1212343456, 1212345432, 1212345456, 1232101010, 1232101210, 1232101212.

Some larger examples :