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zygodromes
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 
25005854850058550 2 
3334441743333646233336462 3 
430035128200234202002342030035130 4 
52002341820023420200234202002342020023420 5 
6167220861666144916661449167220881667501316675013 6 
714302588143024821430226814302302143025511430248214302425 7 
81605063810011710100117101398449113984490100117101001171016050639 8 
9111480941111215411112154111480401111215411112154111480401111215411112154 9 
1010011708100117101001171010011710100117101001171010011710100117101001171010011710 10 
11175828508873686887368688736868873686887368688736868873686887368688736862671074

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.