Saturday, 30 August 2014

Some Interesting Numbers - Kaprekar's Constant, Polygonal Numbers and Highly Composite Numbers

I thought I would wrote a small article on some numbers which I find interesting, I may expand upon this topic in the future, but for this article I'm going to restrict myself to three forms of number: Kaprekar's Constant, Polygonal Numbers and Highly Composite Numbers.

Kaprekar's Constant: 

Kaprekar's Constant is a special constant discovered by the Indian Mathematician called D.R.Kaprekar. The constant has the value of 6174. The constant comes from a simple algorithm known as Kaprekar's Routine. The constant can be produced from at most 7 iterations.

Kaprekar's Constant will always be produced after iterating through Kaprekar's Routine, when given an arbitrary 4 digit integer, providing that at least two of the digits are different otherwise the constant will not be produced.

For example, using 3524 from Wikipedia (since the number of steps is knowingly small), arrange the number in descending order and then ascending order. Subtract these two numbers, and then repeat the process until you reach 6174. You may add any leading 0's to maintain a four digit number.

5432 - 2345 = 3087
8730 - 0378 = 8532
8532 - 2358 = 6174
7641 - 1467 = 6174

Polygonal Numbers:

Polygonal Numbers when arranged as dots will form a polygon like a triangle or square. The Polygonal Numbers usually have a simple formula associated with them.

The first Hexagonal numbers are given as follows:


The general formula for any s-sided polygonal number can be given by the following:

$$P(S,N) = \frac{n^2(s - 2) - n(s-4)}{2}$$

For any given s-sided polygonal number, whereby P(S,N) = X, then the nth term number for X can be found using the following formula:

$$n = \frac{\sqrt{8(s -2) x +(s-4)^2} + (s-4)}{2(s-2)}$$

Highly Composite Numbers:

Highly Composite Numbers are a infinite sequence of numbers with the property, that the number of divisors is greater than any smaller n (any smaller number).

The first Highly Composite Numbers (HCN) are as listed below:

1, 2, 4, 6, 12, 24, 36, 48,...

For example, the number of divisors for 24 is 8, and the all the numbers below 24, have a number of divisors which is not greater than 8 and therefore 24 is considered to be a Highly Composite Number.

There some interesting properties related to Highly Composite Numbers which can be found in the References section.

References:

Highly Composite Number - Wikipedia
Highly Composite Number
Table of Divisors
6174 - Wikipedia
Polygonal Number
Polygonal Number - Wikipedia
 

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