﻿

# 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 $\left(b - 4\right)b^\left\{b - 1\right\} + 2b^\left\{b - 2\right\} + b^\left\{b - 3\right\} + b^4$, 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. [http://mathworld.wolfram.com/Self-DescriptiveNumber.html Self-Descriptive Number] From MathWorld--A Wolfram Web Resource.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Self number — A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer s digits. For example, 21 is not a self number, since it can be generated by… …   Wikipedia

• Descriptive psychology — Psychology …   Wikipedia

• Descriptive Catalogue (1809) — The title page of the Descriptive Catalogue The Descriptive Catalogue of 1809 is a description of, and prospectus for, an exhibition by William Blake of a number of his own illustrations for various topics, but most notably including a set of… …   Wikipedia

• 1000 (number) — List of numbers Integers ← 1k 2k 3k 4k 5k 6k 7k 8k 9k → Cardinal 1000 one thousand …   Wikipedia

• 100000 (number) — List of numbers – Integers 10000 100000 1000000 Cardinal One hundred thousand Ordinal One hundred thousandth Factorization 25 · 55 Roman numeral C Roman numeral (Unicode) …   Wikipedia

• 100 (number) — 100 ← 100 101 102 103 104 105 106 107 108 109 → List of numbers Integers …   Wikipedia

• 1000000000 (number) — List of numbers – Integers 100000000 1000000000 10000000000 Cardinal One billion (short scale) One thousand million (long scale) Ordinal One billionth (short scale) Factorization 29 · 59 Binary 111011100110101100101000000000 He …   Wikipedia

• 100000000 (number) — 100 million redirects here. For the song by Birdman, see 100 Million. One hundred million (100,000,000) is the natural number following 99999999 and preceding 100000001. List of numbers – Integers 10000000 100000000 1000000000 Cardinal One… …   Wikipedia

• 136 (number) — 136 (one hundred [and] thirty six) is the natural number following 135 and preceding 137.Number|number = 136 range = 130s cardinal = one hundred [and] thirty six ordinal = th ordinal text = one hundredth [and] thirty sixth numeral = 136… …   Wikipedia

• Real number — For the real numbers used in descriptive set theory, see Baire space (set theory). For the computing datatype, see Floating point number. A symbol of the set of real numbers …   Wikipedia