repdigits

A number

is a repdigit in a base $b$

it all the digits in its representation in base $b$

are equal.

For example, 7, 222, and 88888 are repdigits in base 10, while $(5555)_7=2000$ is a repdigit in base 7.

In general, a repdigit in base $b>1$ made of $n$ repetitions of the digit $0<d<b$ has value

$d \cdot \frac{b^n-1}{b-1}\,.$

The repdigits composed by repeated ones are called repunits.

A non-trivial repdigit is a number with at least two digits, however it must be noted that every number $n>2$ can be represented as $11$ in base $n-1$ .

The first nontrivial repdigits in base 10 are 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111 more terms

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

A graph displaying how many repdigits are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Repdigit
Wikipedia, Repdigit
OEIS, Sequence A010785

Repdigits can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 aban 11 22 33 + 888 999 abundant 66 88 222 + 999999 6666666 admirable 66 88 222 amenable 33 44 77 + 777777777 888888888 apocalyptic 222 666 2222 + 11111 22222 arithmetic 11 22 33 + 8888888 9999999 binomial 55 66 666 c.decagonal 11 c.heptagonal 22 c.nonagonal 55 Chen 11 congruent 22 55 77 + 8888888 9999999 Cunningham 33 99 999 + 9999999999999 99999999999999 Curzon 33 333333 33333333 cyclic 11 33 77 + 3333333 5555555 D-number 33 111 d-powerful 333 de Polignac 11111111 deceptive 7777 11111 1111111 1111111111 11111111111 deficient 11 22 33 + 8888888 9999999 dig.balanced 11 44 99 + 11111111 66666666 Duffinian 55 77 111 + 1111111 7777777 eban 44 66 economical 11 111 11111 1111111 equidigital 11 111 11111 1111111 Eulerian 11 66 evil 33 66 77 + 777777777 888888888 Fibonacci 55 gapful 1111 2222 3333 + 8888888888 9999999999 good prime 11 happy 44 888 5555 + 1111111 2222222 Harshad 111 222 333 + 888888888 999999999 heptagonal 55 hexagonal 66 hoax 22 1111 6666666 Hogben 111 iban 11 22 44 + 444444 777777 idoneal 22 33 88 insolite 111 111111111 interprime 99 111 4444 Jacobsthal 11 junction 111 11111 22222 + 2222222 22222222 Kaprekar 55 99 999 + 99999999999999 999999999999999 Lehmer 1111 Lucas 11 lucky 33 99 111 + 777777 7777777 magic 111 magnanimous 11 modest 111 222 333 + 999999999 1111111111 Moran 111 222 333 + 888 999 nialpdrome 11 22 33 + 888888888888888 999999999999999 nonagonal 111 6666 nude 11 22 33 + 333333333 444444444 O'Halloran 44 oban 11 33 55 + 888 999 odious 11 22 44 + 99999999 999999999 palindromic 11 22 33 + 888888888888888 999999999999999 palprime 11 pancake 11 22 panconsummate 11 77 pandigital 11 99 partition 11 22 77 pentagonal 22 pernicious 11 22 33 + 5555555 6666666 Perrin 22 plaindrome 11 22 33 + 888888888888888 999999999999999 practical 66 88 666 + 666666 888888 prim.abundant 66 88 222 999999 prime 11 productive 22 66 Proth 33 pseudoperfect 66 88 222 + 888888 999999 repunit 111 1111 11111 + 11111111111111 111111111111111 Ruth-Aaron 77 self 222 1111 88888 + 44444444 555555555 self-describing 22 4444 666666 88888888 semiprime 22 33 55 + 11111 1111111 sliding 11 Smith 22 666 1111 6666666 Sophie Germain 11 sphenic 66 222 555 + 5555555 7777777 straight-line 111 222 333 + 888888888888888 999999999999999 strobogrammatic 11 88 111 + 111111111111111 888888888888888 strong prime 11 subfactorial 44 super-d 333 3333 33333 333333 3333333 tau 88 444 444444444 triangular 55 66 666 tribonacci 44 trimorphic 99 999 9999 + 99999999999999 999999999999999 twin 11 uban 11 22 33 + 88 99 Ulam 11 77 99 + 2222222 9999999 unprimeable 55555 66666 222222 + 6666666 8888888 untouchable 88 66666 222222 upside-down 55 555 5555 + 55555555555555 555555555555555 wasteful 22 33 44 + 8888888 9999999 Woodall 99999999999 Zuckerman 11 111 1111 + 111111111 1111111111 Zumkeller 66 88 222 + 8888 66666 zygodrome 11 22 33 + 888888888888888 999999999999999