A number made of nontrivial runs of identical digits, i.e., such that every digit appear at least in a pair. more
The first 600 zygodromes :
11,
22,
33,
44,
55,
66,
77,
88,
99,
111,
222,
333,
444,
555,
666,
777,
888,
999,
1100,
1111,
1122,
1133,
1144,
1155,
1166,
1177,
1188,
1199,
2200,
2211,
2222,
2233,
2244,
2255,
2266,
2277,
2288,
2299,
3300,
3311,
3322,
3333,
3344,
3355,
3366,
3377,
3388,
3399,
4400,
4411,
4422,
4433,
4444,
4455,
4466,
4477,
4488,
4499,
5500,
5511,
5522,
5533,
5544,
5555,
5566,
5577,
5588,
5599,
6600,
6611,
6622,
6633,
6644,
6655,
6666,
6677,
6688,
6699,
7700,
7711,
7722,
7733,
7744,
7755,
7766,
7777,
7788,
7799,
8800,
8811,
8822,
8833,
8844,
8855,
8866,
8877,
8888,
8899,
9900,
9911,
9922,
9933,
9944,
9955,
9966,
9977,
9988,
9999,
11000,
11100,
11111,
11122,
11133,
11144,
11155,
11166,
11177,
11188,
11199,
11222,
11333,
11444,
11555,
11666,
11777,
11888,
11999,
22000,
22111,
22200,
22211,
22222,
22233,
22244,
22255,
22266,
22277,
22288,
22299,
22333,
22444,
22555,
22666,
22777,
22888,
22999,
33000,
33111,
33222,
33300,
33311,
33322,
33333,
33344,
33355,
33366,
33377,
33388,
33399,
33444,
33555,
33666,
33777,
33888,
33999,
44000,
44111,
44222,
44333,
44400,
44411,
44422,
44433,
44444,
44455,
44466,
44477,
44488,
44499,
44555,
44666,
44777,
44888,
44999,
55000,
55111,
55222,
55333,
55444,
55500,
55511,
55522,
55533,
55544,
55555,
55566,
55577,
55588,
55599,
55666,
55777,
55888,
55999,
66000,
66111,
66222,
66333,
66444,
66555,
66600,
66611,
66622,
66633,
66644,
66655,
66666,
66677,
66688,
66699,
66777,
66888,
66999,
77000,
77111,
77222,
77333,
77444,
77555,
77666,
77700,
77711,
77722,
77733,
77744,
77755,
77766,
77777,
77788,
77799,
77888,
77999,
88000,
88111,
88222,
88333,
88444,
88555,
88666,
88777,
88800,
88811,
88822,
88833,
88844,
88855,
88866,
88877,
88888,
88899,
88999,
99000,
99111,
99222,
99333,
99444,
99555,
99666,
99777,
99888,
99900,
99911,
99922,
99933,
99944,
99955,
99966,
99977,
99988,
99999,
110000,
110011,
110022,
110033,
110044,
110055,
110066,
110077,
110088,
110099,
111000,
111100,
111111,
111122,
111133,
111144,
111155,
111166,
111177,
111188,
111199,
111222,
111333,
111444,
111555,
111666,
111777,
111888,
111999,
112200,
112211,
112222,
112233,
112244,
112255,
112266,
112277,
112288,
112299,
113300,
113311,
113322,
113333,
113344,
113355,
113366,
113377,
113388,
113399,
114400,
114411,
114422,
114433,
114444,
114455,
114466,
114477,
114488,
114499,
115500,
115511,
115522,
115533,
115544,
115555,
115566,
115577,
115588,
115599,
116600,
116611,
116622,
116633,
116644,
116655,
116666,
116677,
116688,
116699,
117700,
117711,
117722,
117733,
117744,
117755,
117766,
117777,
117788,
117799,
118800,
118811,
118822,
118833,
118844,
118855,
118866,
118877,
118888,
118899,
119900,
119911,
119922,
119933,
119944,
119955,
119966,
119977,
119988,
119999,
220000,
220011,
220022,
220033,
220044,
220055,
220066,
220077,
220088,
220099,
221100,
221111,
221122,
221133,
221144,
221155,
221166,
221177,
221188,
221199,
222000,
222111,
222200,
222211,
222222,
222233,
222244,
222255,
222266,
222277,
222288,
222299,
222333,
222444,
222555,
222666,
222777,
222888,
222999,
223300,
223311,
223322,
223333,
223344,
223355,
223366,
223377,
223388,
223399,
224400,
224411,
224422,
224433,
224444,
224455,
224466,
224477,
224488,
224499,
225500,
225511,
225522,
225533,
225544,
225555,
225566,
225577,
225588,
225599,
226600,
226611,
226622,
226633,
226644,
226655,
226666,
226677,
226688,
226699,
227700,
227711,
227722,
227733,
227744,
227755,
227766,
227777,
227788,
227799,
228800,
228811,
228822,
228833,
228844,
228855,
228866,
228877,
228888,
228899,
229900,
229911,
229922,
229933,
229944,
229955,
229966,
229977,
229988,
229999,
330000,
330011,
330022,
330033,
330044,
330055,
330066,
330077,
330088,
330099,
331100,
331111,
331122,
331133,
331144,
331155,
331166,
331177,
331188,
331199,
332200,
332211,
332222,
332233,
332244,
332255,
332266,
332277,
332288,
332299,
333000,
333111,
333222,
333300,
333311,
333322,
333333,
333344,
333355,
333366,
333377,
333388,
333399,
333444,
333555,
333666,
333777,
333888,
333999,
334400,
334411,
334422,
334433,
334444,
334455,
334466,
334477,
334488,
334499,
335500,
335511,
335522,
335533,
335544,
335555,
335566,
335577,
335588,
335599,
336600,
336611,
336622,
336633,
336644,
336655,
336666,
336677,
336688,
336699,
337700,
337711,
337722,
337733,
337744,
337755,
337766,
337777,
337788,
337799,
338800,
338811,
338822,
338833,
338844,
338855,
338866,
338877,
338888,
338899,
339900,
339911,
339922,
339933.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 100117098 values, from 11 to 999999999999999).
n\r | 0 | 1 |
2 | 50058548 | 50058550 | 2 |
3 | 33444174 | 33336462 | 33336462 | 3 |
4 | 30035128 | 20023420 | 20023420 | 30035130 | 4 |
5 | 20023418 | 20023420 | 20023420 | 20023420 | 20023420 | 5 |
6 | 16722086 | 16661449 | 16661449 | 16722088 | 16675013 | 16675013 | 6 |
7 | 14302588 | 14302482 | 14302268 | 14302302 | 14302551 | 14302482 | 14302425 | 7 |
8 | 16050638 | 10011710 | 10011710 | 13984491 | 13984490 | 10011710 | 10011710 | 16050639 | 8 |
9 | 11148094 | 11112154 | 11112154 | 11148040 | 11112154 | 11112154 | 11148040 | 11112154 | 11112154 | 9 |
10 | 10011708 | 10011710 | 10011710 | 10011710 | 10011710 | 10011710 | 10011710 | 10011710 | 10011710 | 10011710 | 10 |
11 | 17582850 | 8873686 | 8873686 | 8873686 | 8873686 | 8873686 | 8873686 | 8873686 | 8873686 | 8873686 | 2671074 |
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.