integer constants. Are there some examples of mathematocal constants which are integer numbers. I know of one that is called Kaprekars constant but thats just a base 10 curiosity. Aret there some more important examples? perhaps in the  fields of combinatorics or abstract algebra? Thanks. It would be optimal if it where 4 digits long.
 A: Integer constants: what do you want?    


*

*The binomial-coeffcients?      

*The Stirling numbers?    

*The Eulerian numbers?    

*The Bell numbers?


Transcendent numbers?


*

*$\pi$ ?

*e (=exp(1))?    


Is that really your question?      

Ok, another try after your comment:      


*

*11 - the first prime p such that the mersenne number $2^p - 1$ is not prime?

*Graham's numbers?          



A: There are a lot of integers in David Well's The Penguin Dictionary of Curious and Interesting Numbers.
A: One fundamental number in geometry is $2$, the ratio between the diameter and the radius of a circle. And therefore also the ratio between the numbers $\tau$ and $\pi$ (or I should say $\tau$ and $\tau/2$). It is also the ratio of the square on the diagonal of a square and the square itslef. It also appears in various other contexts, too much to enumerate here.
A: The number 6 has many interesting properties. The book Lure of the Integers by Joe Roberts lists among them the following:


*

*It is the largest integer which is neither a prime nor the sum of two or more distinct primes

*It is one of only two integers (and the only composite integer) for which $\phi(n) < \sqrt{n}$ (where $\phi$ is Euler's totient function).

*Consider sequences defined by bilinear transformations $x_{n+1} = \frac{ax_n + b}{cx_n + d}$. Given $a,b,c,d,x_0$ such that $\forall n \in \mathbb{N}: x_n \in \mathbb{Z}$, the sequence $x_i$ is periodic with period at most 6.

*Lennes polyhedra of $n$ vertices exist iff $n \ge 6$.

*It is the smallest perfect number.

A: 
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."   -- G. H. Hardy

This is also the only 4-digit number listed on Wikipedia's "Notable integers".
