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 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 
, 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 |
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. The first fact follows trivially from the fact that the digit sum equals the total number of digits, which is equal to the base, from the definition of self-descriptive number.
That a self-descriptive number in base b must be a multiple of that base (or equivalently, that the last digit of the self-descriptive number must be 0) 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 numbers or curious numbers, except that there is no digit length requirement for autobiographical numbers. Self-descriptive numbers are like self numbers only in that they're both base-dependent concepts.
[edit] References
- Clifford Pickover, Keys to Infinity, Chapter 28, "Chaos in Ontario." New York: Wiley, pp. 217-219, 1995.
- Weisstein, Eric W., "Self-Descriptive Number" from MathWorld.
- Sloane's A108551 : Self-descriptive numbers in various bases. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sloane's A046043 : Autobiographical numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
