Numbers of the form abab...a or abab...b. more
The first 600 undulating numbers :
101,
121,
131,
141,
151,
161,
171,
181,
191,
202,
212,
232,
242,
252,
262,
272,
282,
292,
303,
313,
323,
343,
353,
363,
373,
383,
393,
404,
414,
424,
434,
454,
464,
474,
484,
494,
505,
515,
525,
535,
545,
565,
575,
585,
595,
606,
616,
626,
636,
646,
656,
676,
686,
696,
707,
717,
727,
737,
747,
757,
767,
787,
797,
808,
818,
828,
838,
848,
858,
868,
878,
898,
909,
919,
929,
939,
949,
959,
969,
979,
989,
1010,
1212,
1313,
1414,
1515,
1616,
1717,
1818,
1919,
2020,
2121,
2323,
2424,
2525,
2626,
2727,
2828,
2929,
3030,
3131,
3232,
3434,
3535,
3636,
3737,
3838,
3939,
4040,
4141,
4242,
4343,
4545,
4646,
4747,
4848,
4949,
5050,
5151,
5252,
5353,
5454,
5656,
5757,
5858,
5959,
6060,
6161,
6262,
6363,
6464,
6565,
6767,
6868,
6969,
7070,
7171,
7272,
7373,
7474,
7575,
7676,
7878,
7979,
8080,
8181,
8282,
8383,
8484,
8585,
8686,
8787,
8989,
9090,
9191,
9292,
9393,
9494,
9595,
9696,
9797,
9898,
10101,
12121,
13131,
14141,
15151,
16161,
17171,
18181,
19191,
20202,
21212,
23232,
24242,
25252,
26262,
27272,
28282,
29292,
30303,
31313,
32323,
34343,
35353,
36363,
37373,
38383,
39393,
40404,
41414,
42424,
43434,
45454,
46464,
47474,
48484,
49494,
50505,
51515,
52525,
53535,
54545,
56565,
57575,
58585,
59595,
60606,
61616,
62626,
63636,
64646,
65656,
67676,
68686,
69696,
70707,
71717,
72727,
73737,
74747,
75757,
76767,
78787,
79797,
80808,
81818,
82828,
83838,
84848,
85858,
86868,
87878,
89898,
90909,
91919,
92929,
93939,
94949,
95959,
96969,
97979,
98989,
101010,
121212,
131313,
141414,
151515,
161616,
171717,
181818,
191919,
202020,
212121,
232323,
242424,
252525,
262626,
272727,
282828,
292929,
303030,
313131,
323232,
343434,
353535,
363636,
373737,
383838,
393939,
404040,
414141,
424242,
434343,
454545,
464646,
474747,
484848,
494949,
505050,
515151,
525252,
535353,
545454,
565656,
575757,
585858,
595959,
606060,
616161,
626262,
636363,
646464,
656565,
676767,
686868,
696969,
707070,
717171,
727272,
737373,
747474,
757575,
767676,
787878,
797979,
808080,
818181,
828282,
838383,
848484,
858585,
868686,
878787,
898989,
909090,
919191,
929292,
939393,
949494,
959595,
969696,
979797,
989898,
1010101,
1212121,
1313131,
1414141,
1515151,
1616161,
1717171,
1818181,
1919191,
2020202,
2121212,
2323232,
2424242,
2525252,
2626262,
2727272,
2828282,
2929292,
3030303,
3131313,
3232323,
3434343,
3535353,
3636363,
3737373,
3838383,
3939393,
4040404,
4141414,
4242424,
4343434,
4545454,
4646464,
4747474,
4848484,
4949494,
5050505,
5151515,
5252525,
5353535,
5454545,
5656565,
5757575,
5858585,
5959595,
6060606,
6161616,
6262626,
6363636,
6464646,
6565656,
6767676,
6868686,
6969696,
7070707,
7171717,
7272727,
7373737,
7474747,
7575757,
7676767,
7878787,
7979797,
8080808,
8181818,
8282828,
8383838,
8484848,
8585858,
8686868,
8787878,
8989898,
9090909,
9191919,
9292929,
9393939,
9494949,
9595959,
9696969,
9797979,
9898989,
10101010,
12121212,
13131313,
14141414,
15151515,
16161616,
17171717,
18181818,
19191919,
20202020,
21212121,
23232323,
24242424,
25252525,
26262626,
27272727,
28282828,
29292929,
30303030,
31313131,
32323232,
34343434,
35353535,
36363636,
37373737,
38383838,
39393939,
40404040,
41414141,
42424242,
43434343,
45454545,
46464646,
47474747,
48484848,
49494949,
50505050,
51515151,
52525252,
53535353,
54545454,
56565656,
57575757,
58585858,
59595959,
60606060,
61616161,
62626262,
63636363,
64646464,
65656565,
67676767,
68686868,
69696969,
70707070,
71717171,
72727272,
73737373,
74747474,
75757575,
76767676,
78787878,
79797979,
80808080,
81818181,
82828282,
83838383,
84848484,
85858585,
86868686,
87878787,
89898989,
90909090,
91919191,
92929292,
93939393,
94949494,
95959595,
96969696,
97979797,
98989898,
101010101,
121212121,
131313131,
141414141,
151515151,
161616161,
171717171,
181818181,
191919191,
202020202,
212121212,
232323232,
242424242,
252525252,
262626262,
272727272,
282828282,
292929292,
303030303,
313131313,
323232323,
343434343,
353535353,
363636363,
373737373,
383838383,
393939393,
404040404,
414141414,
424242424,
434343434,
454545454,
464646464,
474747474,
484848484,
494949494,
505050505,
515151515,
525252525,
535353535,
545454545,
565656565,
575757575,
585858585,
595959595,
606060606,
616161616,
626262626,
636363636,
646464646,
656565656,
676767676,
686868686,
696969696,
707070707,
717171717,
727272727,
737373737,
747474747,
757575757,
767676767,
787878787,
797979797,
808080808,
818181818,
828282828,
838383838,
848484848,
858585858,
868686868,
878787878,
898989898,
909090909,
919191919,
929292929,
939393939,
949494949,
959595959,
969696969,
979797979,
989898989,
1010101010,
1212121212,
1313131313,
1414141414,
1515151515,
1616161616,
1717171717,
1818181818,
1919191919,
2020202020,
2121212121,
2323232323,
2424242424,
2525252525,
2626262626,
2727272727,
2828282828,
2929292929,
3030303030,
3131313131,
3232323232,
3434343434,
3535353535,
3636363636,
3737373737,
3838383838,
3939393939,
4040404040,
4141414141,
4242424242,
4343434343,
4545454545,
4646464646.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1053 values, from 101 to 989898989898989).
n\r | 0 | 1 |
2 | 498 | 555 | 2 |
3 | 453 | 300 | 300 | 3 |
4 | 246 | 281 | 252 | 274 | 4 |
5 | 165 | 222 | 222 | 222 | 222 | 5 |
6 | 213 | 166 | 151 | 240 | 134 | 149 | 6 |
7 | 295 | 131 | 153 | 116 | 129 | 113 | 116 | 7 |
8 | 123 | 137 | 129 | 137 | 123 | 144 | 123 | 137 | 8 |
9 | 145 | 100 | 100 | 154 | 100 | 100 | 154 | 100 | 100 | 9 |
10 | 54 | 111 | 111 | 111 | 111 | 111 | 111 | 111 | 111 | 111 | 10 |
11 | 56 | 98 | 98 | 98 | 99 | 99 | 99 | 99 | 98 | 98 | 111 |
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.