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, 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)bb − 1 + 2bb − 2 + bb − 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.
