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digitally balanced numbers
A number is digitally balanced in base  $b$  if it contains all the digits  $0,1,\dots,(b-1)$, an equal number of times.

For example, 1023456789 is balanced in base 10, while  $3526664=(1400323124)_5$  is balanced in base 5.

The smallest 3 × 3 magic square made of consecutive balanced numbers in any base is

(11101001001100)2 (11101001000101)2 (11101001001010)2
(202110120)3 (11101001001001)2 (202110201)3
(153024)6 (202110210)3 (11101001000110)2

which corresponds to the 9 consecutive numbers 14924, 14917, 14922, 14919, 14921, 14923, 14920, 14925, and 14918.

The first numbers balanced in any base are 2, 9, 10, 11, 12, 15, 19, 21, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 75, 78, 99, 108, 114, 120, 135 more terms

Below, the spiral pattern of digitally balanced numbers (in any base) up to 10000. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of balanced numbers

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

A graph displaying how many digitally balanced numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of digitally balanced numbers
Useful links OEIS, Sequence A031443
Digitally balanced numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 3221 8543 + 199971077 ABA 50 384 450 + 199520288 aban 10 11 12 + 199000996 abundant 12 42 56 + 49999968 Achilles 108 500 675 + 199895904 admirable 12 42 56 + 95510716 alt.fact. 19 35899 36614981 alternating 10 12 21 + 189898989 amenable 12 21 37 + 199999936 amicable 1210 2620 12285 + 196421715 apocalyptic 226 278 312 + 29965 arithmetic 11 15 19 + 9999996 astonishing 15 204 216 + 23490 automorphic 625 balanced p. 563 653 2417 + 199987841 Bell 15 52 678570 190899322 bemirp 16001 198901 11816011 + 189166801 betrothed 75 195 8892 + 184191111 binomial 10 15 21 + 199850028 brilliant 10 15 21 + 199983713 c.decagonal 11 551 661 + 197160601 c.heptagonal 197 316 841 + 199852423 c.nonagonal 10 595 820 + 199690120 c.octagonal 49 169 225 + 199967881 c.pentagonal 141 226 601 + 199741956 c.square 41 613 841 + 199460365 c.triangular 10 19 166 + 199843960 cake 15 42 232 + 199065882 Carmichael 10585 15841 162401 + 193910977 Carol 16127 Catalan 42 58786 35357670 Chen 11 19 37 + 67084289 compositorial 2903040 congruent 15 21 37 + 9999996 constructible 10 12 15 + 178257920 cube 216 2744 15625 + 187149248 Cullen 10241 Cunningham 10 15 35 + 199967882 Curzon 21 41 50 + 199999749 cyclic 11 15 19 + 9999989 D-number 15 21 141 + 7043079 d-powerful 135 153 209 + 9999827 de Polignac 149 809 905 + 67078145 decagonal 10 52 232 + 199692226 deceptive 11111 12403 14911 + 199374721 deficient 10 11 15 + 9999994 double fact. 15 384 10395 droll 240 247808 Duffinian 21 35 49 + 9999993 eban 42 44 50 + 66066062 economical 10 11 15 + 16771072 emirp 37 149 709 + 199999777 emirpimes 15 49 169 + 66996737 enlightened 2744 236196 250000 equidigital 10 11 15 + 16771072 eRAP 170 2079 197110 + 199977000 esthetic 10 12 21 + 65676565 Eulerian 11 120 302 + 134217700 evil 10 12 15 + 199999936 factorial 120 3628800 fibodiv 19 75 149 + 199546494 Fibonacci 21 14930352 Friedman 153 216 625 + 995463 frugal 625 3125 3721 + 199999501 gapful 108 120 135 + 199999936 Gilda 49 78 550 + 14005576 good prime 11 37 41 + 199784461 happy 10 19 44 + 9999919 harmonic 27846 55860 167400 + 191711520 Harshad 10 12 21 + 199999936 heptagonal 970 2205 2673 + 199840291 hex 19 37 169 + 199389769 hexagonal 15 120 153 + 199310595 highly composite 12 120 180 + 14414400 hoax 166 202 308 + 95581084 Hogben 21 601 651 + 199812361 Honaker 2141 2221 2707 + 199805033 house 78 434 652 + 189512709 hungry 712201 hyperperfect 21 10693 163201 + 194283181 iban 10 11 12 + 777744 iccanobiF 12815 792517 idoneal 10 12 15 + 520 inconsummate 75 195 216 + 999993 insolite 11112 interprime 12 15 21 + 95583996 Jacobsthal 11 21 699051 Jordan-Polya 12 120 216 + 8847360 junction 202 204 210 + 95584540 Kaprekar 99 500500 643357 + 55474452 katadrome 10 21 41 + 9876540 Lehmer 15 595 2701 + 199803151 Leyland 177 2169 2530 + 10609137 lonely 120 15705 31428 + 142414669 Lucas 11 167761 lucky 15 21 37 + 9999963 Lynch-Bell 12 15 135 + 9812376 m-pointer 15121 2111411 2121131 + 163111111 magic 15 260 8801 + 199344496 magnanimous 11 12 21 + 62202281 metadrome 12 15 19 + 12345689 modest 19 49 209 + 199890243 Moran 21 42 114 + 95454108 Motzkin 21 835 41835 narcissistic 153 1634 146511208 nialpdrome 10 11 21 + 88855500 nonagonal 75 154 204 + 199648449 nude 11 12 15 + 199996128 O'Halloran 12 44 156 + 660 oban 10 11 12 + 990 octagonal 21 225 936 + 199985345 odious 11 19 21 + 199999932 Ormiston 18397 34613 34631 + 199972873 palindromic 11 44 99 + 199505991 palprime 11 787 929 + 199393991 pancake 11 37 56 + 199830037 panconsummate 10 11 12 + 721 pandigital 11 15 19 + 199999932 partition 11 15 42 + 169229875 pentagonal 12 35 210 + 199890132 pernicious 10 11 12 + 9999925 Perrin 10 12 209 + 727653 Pierpont 19 37 163 plaindrome 11 12 15 + 188899999 Poulet 2701 7957 8321 + 197747377 power 49 169 216 + 199967881 powerful 49 108 169 + 199967881 practical 12 42 56 + 9999990 prim.abundant 12 42 56 + 95510716 prime 11 19 37 + 199999777 primeval 37 10136 10279 + 10234579 primorial 210 pronic 12 42 56 + 199586256 Proth 41 49 177 + 67092481 pseudoperfect 12 42 56 + 999996 repdigit 11 44 99 + 66666666 repfigit 19 75 197 + 298320 repunit 15 21 156 + 199812361 Rhonda 2835 15625 25662 + 197521734 Ruth-Aaron 15 49 50 + 199854851 Saint-Exupery 14760 20580 30720 + 199977120 Sastry 528 13224 147928995 178838403 self 42 75 108 + 199999874 self-describing 10123331 10143331 10173133 + 42292924 semiprime 10 15 21 + 67092481 sliding 11 52 290 + 195824500 Smith 166 202 438 + 95580940 Sophie Germain 11 41 653 + 199993361 sphenic 42 78 114 + 67078146 square 49 169 225 + 199967881 star 37 541 661 + 199861273 straight-line 135 147 210 + 66666666 strobogrammatic 11 689 818 + 199000661 strong prime 11 37 41 + 67084289 subfactorial 44 super Niven 10 12 50 + 180000000 super-d 19 169 318 + 9999981 superabundant 12 120 180 + 10810800 tau 12 56 108 + 199999784 taxicab 134379 149389 195841 + 191213568 tetrahedral 10 35 56 + 196821856 tetranacci 15 56 108 + 54114452 triangular 10 15 21 + 199850028 tribonacci 44 149 10609 + 8646064 trimorphic 49 75 99 + 50000001 truncatable prime 37 197 613 + 198739397 twin 11 19 41 + 199994369 uban 10 11 12 + 69000020 Ulam 11 38 99 + 10000241 undulating 141 202 212 + 181818181 unprimeable 204 518 532 + 9999982 untouchable 52 120 210 + 999996 upside-down 19 37 555 + 199852119 vampire 136948 145314 152685 + 89598460 wasteful 12 38 42 + 9999996 weak prime 19 139 601 + 67076353 weakly prime 8353427 9537371 14075273 + 196363373 weird 10570 12530 13370 + 997570 Wieferich 14209 45643 157995 + 684645 Woodall 2499 52487 705893 10485759 Zeisel 25085 6173179 8413179 + 185245273 Zuckerman 11 12 15 + 198731232 Zumkeller 12 42 56 + 65280 zygodrome 11 44 99 + 118800444