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 |
2 | 79821 | 82083 | 2 |
3 | 54007 | 53944 | 53953 | 3 |
4 | 38149 | 42736 | 41672 | 39347 | 4 |
5 | 29663 | 33677 | 35135 | 34008 | 29421 | 5 |
6 | 26625 | 27349 | 26601 | 27382 | 26595 | 27352 | 6 |
7 | 23145 | 23115 | 23137 | 23123 | 23137 | 23134 | 23113 | 7 |
8 | 22572 | 21195 | 20506 | 15971 | 15577 | 21541 | 21166 | 23376 | 8 |
9 | 17992 | 17978 | 17984 | 17999 | 17992 | 17979 | 18016 | 17974 | 17990 | 9 |
10 | 6287 | 12511 | 16983 | 21195 | 22572 | 23376 | 21166 | 18152 | 12813 | 6849 | 10 |
11 | 8543 | 19668 | 12924 | 19643 | 16492 | 16987 | 16021 | 11590 | 15063 | 7618 | 17355 |
A pictorial representation of the table above
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.