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esthetic numbers
A number in which adjacent digits differ by 1. more

The first 600 esthetic numbers :
10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876, 878, 898, 987, 989, 1010, 1012, 1210, 1212, 1232, 1234, 2101, 2121, 2123, 2321, 2323, 2343, 2345, 3210, 3212, 3232, 3234, 3432, 3434, 3454, 3456, 4321, 4323, 4343, 4345, 4543, 4545, 4565, 4567, 5432, 5434, 5454, 5456, 5654, 5656, 5676, 5678, 6543, 6545, 6565, 6567, 6765, 6767, 6787, 6789, 7654, 7656, 7676, 7678, 7876, 7878, 7898, 8765, 8767, 8787, 8789, 8987, 8989, 9876, 9878, 9898, 10101, 10121, 10123, 12101, 12121, 12123, 12321, 12323, 12343, 12345, 21010, 21012, 21210, 21212, 21232, 21234, 23210, 23212, 23232, 23234, 23432, 23434, 23454, 23456, 32101, 32121, 32123, 32321, 32323, 32343, 32345, 34321, 34323, 34343, 34345, 34543, 34545, 34565, 34567, 43210, 43212, 43232, 43234, 43432, 43434, 43454, 43456, 45432, 45434, 45454, 45456, 45654, 45656, 45676, 45678, 54321, 54323, 54343, 54345, 54543, 54545, 54565, 54567, 56543, 56545, 56565, 56567, 56765, 56767, 56787, 56789, 65432, 65434, 65454, 65456, 65654, 65656, 65676, 65678, 67654, 67656, 67676, 67678, 67876, 67878, 67898, 76543, 76545, 76565, 76567, 76765, 76767, 76787, 76789, 78765, 78767, 78787, 78789, 78987, 78989, 87654, 87656, 87676, 87678, 87876, 87878, 87898, 89876, 89878, 89898, 98765, 98767, 98787, 98789, 98987, 98989, 101010, 101012, 101210, 101212, 101232, 101234, 121010, 121012, 121210, 121212, 121232, 121234, 123210, 123212, 123232, 123234, 123432, 123434, 123454, 123456, 210101, 210121, 210123, 212101, 212121, 212123, 212321, 212323, 212343, 212345, 232101, 232121, 232123, 232321, 232323, 232343, 232345, 234321, 234323, 234343, 234345, 234543, 234545, 234565, 234567, 321010, 321012, 321210, 321212, 321232, 321234, 323210, 323212, 323232, 323234, 323432, 323434, 323454, 323456, 343210, 343212, 343232, 343234, 343432, 343434, 343454, 343456, 345432, 345434, 345454, 345456, 345654, 345656, 345676, 345678, 432101, 432121, 432123, 432321, 432323, 432343, 432345, 434321, 434323, 434343, 434345, 434543, 434545, 434565, 434567, 454321, 454323, 454343, 454345, 454543, 454545, 454565, 454567, 456543, 456545, 456565, 456567, 456765, 456767, 456787, 456789, 543210, 543212, 543232, 543234, 543432, 543434, 543454, 543456, 545432, 545434, 545454, 545456, 545654, 545656, 545676, 545678, 565432, 565434, 565454, 565456, 565654, 565656, 565676, 565678, 567654, 567656, 567676, 567678, 567876, 567878, 567898, 654321, 654323, 654343, 654345, 654543, 654545, 654565, 654567, 656543, 656545, 656565, 656567, 656765, 656767, 656787, 656789, 676543, 676545, 676565, 676567, 676765, 676767, 676787, 676789, 678765, 678767, 678787, 678789, 678987, 678989, 765432, 765434, 765454, 765456, 765654, 765656, 765676, 765678, 767654, 767656, 767676, 767678, 767876, 767878, 767898, 787654, 787656, 787676, 787678, 787876, 787878, 787898, 789876, 789878, 789898, 876543, 876545, 876565, 876567, 876765, 876767, 876787, 876789, 878765, 878767, 878787, 878789, 878987, 878989, 898765, 898767, 898787, 898789, 898987, 898989, 987654, 987656, 987676, 987678, 987876, 987878, 987898, 989876, 989878, 989898, 1010101, 1010121, 1010123, 1012101, 1012121, 1012123, 1012321, 1012323, 1012343, 1012345, 1210101, 1210121, 1210123, 1212101, 1212121, 1212123, 1212321, 1212323, 1212343, 1212345, 1232101, 1232121, 1232123, 1232321, 1232323, 1232343, 1232345, 1234321, 1234323, 1234343, 1234345, 1234543, 1234545, 1234565, 1234567, 2101010, 2101012, 2101210, 2101212, 2101232, 2101234, 2121010, 2121012, 2121210, 2121212, 2121232, 2121234, 2123210, 2123212, 2123232, 2123234, 2123432, 2123434, 2123454, 2123456, 2321010, 2321012, 2321210, 2321212, 2321232, 2321234, 2323210, 2323212, 2323232, 2323234, 2323432, 2323434, 2323454, 2323456, 2343210, 2343212, 2343232, 2343234, 2343432, 2343434, 2343454, 2343456, 2345432, 2345434, 2345454, 2345456, 2345654, 2345656, 2345676, 2345678, 3210101, 3210121, 3210123, 3212101, 3212121, 3212123, 3212321, 3212323, 3212343, 3212345, 3232101, 3232121, 3232123, 3232321, 3232323, 3232343, 3232345, 3234321, 3234323, 3234343, 3234345, 3234543, 3234545, 3234565, 3234567, 3432101, 3432121, 3432123, 3432321, 3432323, 3432343, 3432345, 3434321, 3434323, 3434343, 3434345, 3434543, 3434545, 3434565, 3434567, 3454321, 3454323, 3454343, 3454345, 3454543, 3454545, 3454565, 3454567, 3456543, 3456545, 3456565, 3456567, 3456765, 3456767, 3456787, 3456789, 4321010, 4321012, 4321210, 4321212, 4321232, 4321234, 4323210, 4323212, 4323232, 4323234, 4323432.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 161904 values, from 10 to 989898989898989).

n\r 0  1 
27982182083 2 
3540075394453953 3 
438149427364167239347 4 
52966333677351353400829421 5 
6266252734926601273822659527352 6 
723145231152313723123231372313423113 7 
82257221195205061597115577215412116623376 8 
9179921797817984179991799217979180161797417990 9 
10628712511169832119522572233762116618152128136849 10 
1185431966812924196431649216987160211159015063761817355

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.