Self-descriptive number

From Wikipedia, the free encyclopedia.

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 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 , 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 (Ellipses omit 23 zeroes)	Approx. 2.14349 × 10⁵³

Sloane's (sequence 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 (sequence in OEIS) lists base 10 autobiographical numbers). Self-descriptive numbers are like self numbers only in that they're both base-dependent concepts.

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.