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 of the following reasons:

In base 10, the number has 10 digits;
It contains 6 at position 0, indicating that there are six 0s;
It contains 2 at position 1, indicating that there are two 1s;
It contains 1 at position 2, indicating that there is one 2;
It contains 0 at position 3, indicating that there is no 3;
It contains 0 at position 4, indicating that there is no 4;
It contains 0 at position 5, indicating that there is no 5;
It contains 1 at position 6, indicating that there is one 6;
It contains 0 at position 7, indicating that there is no 7;
It contains 0 at position 8, indicating that there is no 8;
It contains 0 at position 9, indicating that there is no 9.

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} + b⁴, 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 (sequence A108551 in OEIS) 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 A046043 lists base 10 autobiographical numbers). Self-descriptive numbers are like self numbers only in that they're both base-dependent concepts.

[edit] External references

• 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.