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self-describing numbers
A number  $n$  is called self-describing if it has an even number of digits, so that the digits can be divided into adjacent pairs  $n=q_1d_1\,q_2d_2\,\dots\,q_kd_k$  and pair  $q_id_i$  truthfully declares that the number  $n$  contains  $q_i$  copies of digit  $d_i$.

All digits must be accounted for, but pairs can be repeated.

For example, the number  $31101233$  is divided into the pairs  $31$,  $10$,  $12$,  $33$, this say: the number contains three  $1$, one  $0$, one  $2$  and three  $3$. Another example, the number  $4444$, divided into  $44$,  $44$, tells us (twice) that it contains four  $4$.

The self-describing numbers are not very common. Up to  $10^{15}$  (actually up to  $10^{14}$, since they must have an even number of digits) there are 783343 such numbers.

The smallest pandigital one is 10141516181923273271.

The self-describing numbers are finite, since we can have at most 9 copies of each digit. According to Robert G. Wilson the last term could be

\[\small 9998979595959595848484848484848476737373737373736262626262625151515110.\]

The first self-describing numbers are 22, 4444, 224444, 442244, 444422, 666666, 10123133, 10123331, 10143133, 10143331, 10153133 more terms

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

a-pointer 32162321 + 3316221431 4116104441 4144151741 4144171541 aban 22 abundant 442244 + 42212124 42242020 42262624 42272724 admirable 14313318 + 32232116 33153114 33183114 33311514 amenable 4444 + 42282824 42292429 42292924 88888888 arithmetic 22 4444 224444 444422 666666 balanced p. 17331031 + 1814223133 1833221431 4441151641 4442274227 brilliant 12193331 + 24294229 31121533 31121933 31151933 c.heptagonal 22 Chen 10153331 + 33151931 33161231 33311417 33322727 congruent 22 4444 224444 444422 666666 Curzon 10183133 + 33293229 33311418 33311714 42252425 de Polignac 10143133 + 33101831 33121631 33141731 33193117 deficient 22 4444 224444 444422 dig.balanced 10123331 + 42202024 42202420 42282428 42292924 economical 10123133 + 19331231 19331731 19331831 19333115 emirp 10233221 + 31121833 31183319 32102321 33322727 emirpimes 10183331 + 33311219 33311419 33311617 33311819 equidigital 10123133 + 19331231 19331731 19331831 19333115 evil 4444 + 42262624 42292429 66662266 88888888 frugal 10311233 19322123 31331417 33311918 gapful 4444 + 4466666644 8822888888 8888228888 8888882288 good prime 12103331 Harshad 10313316 + 4441411712 4441411912 6666446644 8822888888 hex 22154133413419 hoax 22 + 32263326 33161431 33173115 33322828 Honaker 18331031 19143133 19143331 21183223 iban 22 4444 224444 442244 444422 idoneal 22 inconsummate 224444 interprime 4444 + 33311714 33311814 42272427 42272724 junction 444422 + 42272427 42272724 42282428 42292429 Kaprekar 88888888 modest 4444 + 22666666 24422727 66226666 88888888 Moran 12173133 + 31331214 32152123 33103119 33253225 nialpdrome 22 + 666666444422 888888884444 88888888444422 88888888666666 nonagonal 1041184144 12351833511751 nude 22 + 33322626 42212124 42282824 88888888 odious 22 + 42282824 42292924 66226666 66666622 Ormiston 18103331 1422331931 1733221531 1822143331 1910223331 palindromic 22 + 88448822884488 88668866886688 88884422448888 88886666668888 pancake 22 partition 22 pentagonal 22 pernicious 22 224444 442244 444422 Perrin 22 plaindrome 22 + 224444666666 444488888888 22444488888888 66666688888888 Poulet 33193117 practical 666666 prim.abundant 14313318 + 32232116 33153114 33183114 33311514 prime 10153331 + 622927292627 622928282629 622929262727 622929272627 pseudoperfect 442244 666666 repdigit 22 4444 666666 88888888 self 666666 + 33322727 33322828 33322929 42292924 semiprime 22 + 33311917 42232423 42242929 42292429 Smith 22 + 33161431 33173115 33292932 42272724 sphenic 444422 + 42242626 42242727 42262426 42272427 straight-line 4444 666666 88888888 strobogrammatic 88888888 strong prime 10153331 + 33181231 33311017 33311417 33322727 super-d 444422 tau 10313316 + 33311016 33311712 33311916 33322424 twin 10153331 + 32212319 33141931 33151231 33151931 uban 22 wasteful 22 + 224444 442244 444422 666666 weak prime 10233221 + 33141931 33151231 33161831 33311519 Zuckerman 14333112 31143312 zygodrome 22 + 88888888224444 88888888442244 88888888444422 88888888666666