SELF-DESCRIPTIVE NUMBER

A 'self-descriptive number' is an integer ''m'' that in a given base ''b'' is ''b''-digits long in which each digit ''d'' at position ''n'' (the most significant digit being at position 0 and the least significant at position ''b'' - 1) counts how many instances of digit ''n'' are in ''m''.
For example, in base 10, the number 6210001000 is self-descriptive because it has six 0s, two 1s, one 2, one 6, and no 3s, 4s, 5s, 7s, 8s or 9s.
There are no self-descriptive numbers in bases 2, 3 or 6. In bases 7 and above, there is, if nothing else, a self-descriptive number of the form $(b\; -\; 4)b^\{b\; -\; 1\}\; +\; 2b^\{b\; -\; 2\}\; +\; b^\{b\; -\; 3\}\; +\; b^4$, which has ''b'' - 4 instances of the digit 0, two instances of the digit 1, one instance of the digit 2, one instance of digit ''b'' - 4, and no instances of any other digits. The following table lists some self-descriptive numbers in a few selected bases:

Base	Self-descriptive numbers	Values in base 10
4	1210, 2020	100, 136
5	21200	1425
7	3211000	389305
8	42101000	8946176
9	521001000	225331713
10	6210001000	6210001000
16	C210000000001000	13983676842985394176
36	W21000 ... 0001000 (Ellipsis omits 23 zeroes)	Approx. 2.14349 × 1053

Sloane's lists a few more self-descriptive numbers.
From the numbers listed in the table, it would seem that all self-descriptive numbers have digit sums equal to their base, and that they're multiples of that base.
That a self-descriptive number in base ''b'' must be a multiple of that base can be proven ad absurda as follows: assume that there is in fact a self-descriptive number ''m'' in base ''b'' that is ''b''-digits long but not a multiple of ''b''. The digit at position ''b'' - 1 must be at least 1, meaning that there is at least one instance of the digit ''b'' - 1 in ''m''. At whatever position ''x'' that digit ''b'' - 1 falls, there must be at least ''b'' - 1 instances of digit ''x'' in ''m''. Therefore, we have at least one instance of the digit 1, and ''b'' - 1 instances of ''x''. If ''x'' > 1, then ''m'' has more than ''b'' digits, leading to a contradiction of our initial statement. And if ''x'' = 0 or 1, that also leads to a contradiction.
The concept of self-descriptive numbers is similar to that of autobiographical or curious numbers, except that there is no digit length requirement for autobiographical numbers. (Sloane's lists base 10 autobiographical numbers). Self-descriptive numbers are like self numbers only in that they're both base-dependent concepts.

Contents

External references

External references

★ Alexander Bogomolny Self-descriptive Strings

★ Alexander Bogomolny Self-documenting Sentences

★ Clifford Pickover, ''Keys to Infinity'', Chapter 28, "Chaos in Ontario." New York: Wiley, pp. 217-219, 1995.

★ Eric W. Weisstein. Self-Descriptive Number From MathWorld--A Wolfram Web Resource.