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:

Sloane's OEIS|id=A108551 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 OEIS2C|id=A046043 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.

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