Friday, 23 February 2007

What's Special About This Number?

via BoingBoing

BoingBoing listed the first 11 numbers. Here is the next 10.

11 is the largest known multiplicative persistence.
12 is the smallest abundant number.
13 is the number of Archimedian solids.
14 is the smallest number n with the property that there are no numbers relatively prime to n smaller numbers.
15 is the smallest composite number n with the property that there is only one group of order n.
16 is the only number of the form xy = yx with x and y different integers.
17 is the number of wallpaper groups.
18 is the only number that is twice the sum of its digits.
19 is the maximum number of 4th powers needed to sum to any number.
20 is the number of rooted trees with 6 vertices.

Frankly, I don't a lot of the special properties, e.g.

Abundant Number [] is a positive integer n for which

where sigma(n) is the divisor function and s(n) is the restricted divisor function. The quantity sigma(n)-2n is sometimes called the abundance. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (Sloane's A005101). Abundant numbers are sometimes called excessive numbers.

There are only 21 abundant numbers less than 100, and they are all even. The first odd abundant number is

That 945 is abundant can be seen by computing

Any multiple of a perfect number or an abundant number is also abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.

Mathematically oriented people surely will be interested in the whole list (from 0 to 9999) which only a few numbers without some significance.

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