User blog:ArtismScrub/Defining a constant, maybe?

This series of constants is based on the factorial.

It can be shown that the factorial can be bounded at n! < nn-1, by simple logic:

n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1, which contains n terms in total

=  n * (n-1) * (n-2) * ... * 3 * 2, which contains n-1 terms in total (the 1 at the end does nothing)

< n * n * n * ... * n * n * n where there are n-1 terms in total

=  n n-1

Some people give  n n  as the ideal bound, but since n*1=n, that term can be ignored.

In fact, since exponentiation has been generalized to allow any real numbers (or imaginary/complex, for that matter), it can be possible to give specific values of n! = nc for specific n, using the formula:

cn = logn(n!)

The first few values of cn (calculated here ) are:

1! = 1 = 1(literally anything)

2! = 2 = 21

3! = 6 ≈ 31.6309297536

4! = 24 ≈ 42.2924812504

5! = 120  ≈ 52.9746358687

6! = 720  ≈ 63.6719500162

7! = 5040  ≈  74.3810662919

8! = 40320  ≈ 85.0997360061

9! = 362880  ≈ 95.8263627724

10! = 3628800  ≈ 106.5597630328767937511747612399601 (this one was calculated using Windows 7 calc instead because it supports the common logarithm)

...

25! = 15511210043330985984000000  ≈ 2518.019833128

etc.

When I was initially writing this, my thought was that (n - cn) would converge to a finite value as n approached infinity... that doesn't seem to be the case...

So, how about values of n/cn? (calculated with Windows 7 calculator)

1/c1 = undefined

2/c2 = 2

3/c3  ≈   1.8394415782641835543492533656601

4/c 4  ≈  1.7448343358542479968629190756761

5/c5  ≈  1.6808780034596777065347433662511

6/c6  ≈  1.6340091704759191814404088456176

7/c7  ≈  1.5977845423024195713837059549197

8/c8  ≈  1.5687086528461232537603217405885

9/c9  ≈  1.5447029907979301505259631539795

10/c10  ≈  1.5244453114969437019248437622015

...

25/c25  ≈  1.3873602392662512118686030486974

Now, define χ as the limit of n/cn as n approaches infinity.

Worst case scenario, χ will just be equal to 1.