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zygodromes
A number is a zygodrome in a given base  $b$  (usually 10) if in that base it is made of nontrivial runs of identical digits.

I call these numbers zygodromes because digits appear at least in pairs and the Greek root for pair is zygo-.

For example, 112277, 44444333 and 55500111 are all zygodromes in base 10.

The total number  $c(n)$  of zygodromes in base 10 with at most  $n$  digits is given by the recurrence defined by  $c(1)=0$,  $c(2)=9$,  $c(3)=18$  and

\[ c(n)=2\cdot c(n - 1) + 8\cdot c(n - 2) - 9\cdot c(n - 3)\,. \]

The sum of the reciprocals of zygodromes converges to a number approximatively equal to 0.3164971.

The prime  $p_{664444}=9997777$  is the only known zygodrome prime whose index is a zygodrome as well.

The first zygodromes (in base 10) are 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1100, 1111, 1122, 1133 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of zygodromes

A graph displaying how many zygodromes are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of zygodromes
Useful links OEIS, Sequence A033023
Zygodromes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 4443311 33322111 + 9922288811 ABA 447722888 11000004488 22244466888 + 8877886668800 aban 11 22 33 + 999999000999 abundant 66 88 222 + 44999988 Achilles 665500 5556600 6655000 + 44885588800000 admirable 66 88 222 + 99933222 amenable 33 44 77 + 999999988 apocalyptic 222 666 1133 + 22999 arithmetic 11 22 33 + 9999999 astonishing 3388 balanced p. 3355577 5555777 7722277 + 9993355511 binomial 55 66 666 + 5553333000111 brilliant 22999 55577 1166111 + 999997777 c.decagonal 11 22111 445511 + 884441155522111 c.heptagonal 22 4411 669922 + 11662266115522 c.nonagonal 55 111221155 c.triangular 6644778888844 8882229992266 55777555332244 + 110055449944444 Chen 11 11177 11777 + 99955577 congruent 22 55 77 + 9999999 Cunningham 33 99 999 + 557755556688800 Curzon 33 1133 1166 + 119999333 cyclic 11 33 77 + 9999977 D-number 33 111 11199 + 6666999 d-powerful 333 2244 22333 + 9933222 de Polignac 1199 7799 44111 + 99996677 decagonal 1117777 22994422 6663334455 + 880000077224422 deceptive 7777 11111 1111111 + 11188888777 deficient 11 22 33 + 9999999 dig.balanced 11 44 99 + 118800444 Duffinian 55 77 111 + 9999977 eban 44 66 44000 + 66000066000066 economical 11 111 6655 + 11999933 emirp 11777 77711 1133333 + 119988899 emirpimes 1177 7711 7799 + 99999911 enlightened 119911 equidigital 11 111 6655 + 11999933 eRAP 2255 66661111 336655777 Eulerian 11 66 evil 33 66 77 + 999999988 fibodiv 114488 Fibonacci 55 Friedman 116655 117755 117777 + 449955 frugal 1100000 1113344 1118866 + 999666111 gapful 1100 1111 1155 + 99999999900 good prime 11 11777 33311 + 117770011 happy 44 888 1122 + 9998877 Harshad 111 222 333 + 9999999666 heptagonal 55 1177 11122 + 881118800088111 hex 1188811 2200777 661166611 + 552299998822111 hexagonal 66 1188111 2200066611 + 558822550004400 highly composite 554400 hoax 22 1111 3344 + 99991155 Hogben 111 33777110011 77333770011 + 2255660055111 Honaker 22277 77711 1144499 + 999112211 iban 11 22 44 + 777777 idoneal 22 33 88 inconsummate 1166 3311 4466 + 999933 insolite 111 111111111 interprime 99 111 1100 + 99996633 Jacobsthal 11 junction 111 3322 4422 + 99999966 Kaprekar 55 99 999 + 999999999999999 Lehmer 1111 4411 11155 + 887755665511 Lucas 11 lucky 33 99 111 + 9999555 magic 111 665555 665500005500 665500000055000 magnanimous 11 modest 111 222 333 + 1199999999 Moran 111 222 333 + 99966699 nialpdrome 11 22 33 + 999999999999999 nonagonal 111 5500 6666 + 990011222550099 nude 11 22 33 + 449996688 O'Halloran 44 oban 11 33 55 + 999 octagonal 1133002200 113300220033 2222669966000 + 222266996600033 odious 11 22 44 + 999999999 palindromic 11 22 33 + 999999999999999 palprime 11 1177711 7722277 + 999955444559999 pancake 11 22 1177 + 881155771115566 panconsummate 11 77 pandigital 11 99 33555 + 339955500 partition 11 22 77 pentagonal 22 8855 55777 + 88844444113322 pernicious 11 22 33 + 9999988 Perrin 22 plaindrome 11 22 33 + 999999999999999 power 7744 774400 77440000 + 8844449977444 powerful 7744 665500 774400 + 992255544007744 practical 66 88 666 + 9999900 prim.abundant 66 88 222 + 99933222 prime 11 11177 11777 + 999999988811 pronic 1122 4422 9900 + 99999990000000 Proth 33 8833 11777 + 222554488833 pseudoperfect 66 88 222 + 999999 repdigit 11 22 33 + 999999999999999 repunit 111 1111 11111 + 111111111111111 Rhonda 333355511122 Ruth-Aaron 77 2299 8855 + 449990004477 Saint-Exupery 555660000 555660000000 554444445555000 555660000000000 self 222 1111 1122 + 999999555 self-describing 22 4444 224444 + 88888888666666 semiprime 22 33 55 + 99999911 sliding 11 1100 11000 + 110000000000000 Smith 22 666 1111 + 99991155 Sophie Germain 11 44111 77711 + 9999988811 sphenic 66 222 555 + 99999922 square 7744 774400 77440000 + 992255544007744 star 220033 771133 332255533 77111779933 straight-line 111 222 333 + 999999999999999 strobogrammatic 11 88 111 + 999999888666666 strong prime 11 11777 22277 + 99955577 subfactorial 44 super Niven 1100 2200 3300 + 44880000000 super-d 333 1188 3322 + 9999666 superabundant 554400 tau 88 444 22000 + 999992244 tetrahedral 55228866 1188887700 triangular 55 66 666 + 558822550004400 tribonacci 44 trimorphic 99 999 9999 + 999999999999999 twin 11 11777 22111 + 999922211 uban 11 22 33 + 99000099000099 Ulam 11 77 99 + 9999999 unprimeable 1144 5544 6622 + 9999988 untouchable 88 3366 3388 + 999966 upside-down 55 555 1199 + 999999555111111 vampire 44995500 2266335544 5522776600 wasteful 22 33 44 + 9999999 weak prime 11177 22111 44777 + 99955111 weakly prime 77755114477 weird 776644 Woodall 99999999999 11999999999999 Zeisel 336611 Zuckerman 11 111 1111 + 3311331111 Zumkeller 66 88 222 + 99988