# Number of ways a natural number can be written as sum of smaller natural number

It is easy to realize that given a natural number N the number of doublets that sum N are $\frac{N+(-1)(N \pmod 2)}{2}$ , so I thought I could reach some recursive formula in the sense that found the number of doublets I could find the number of triplets and so on ..., example: N=3 the only doublet is 2+1=3 -not said yet but 2+1, and 1+2 count as one- then I could count the number of way the number 2 can be expressed as the indicated sum and got the total number of ways 3 can be written as a sum. But this seems not so efficient, so I was wondering if there is other way to attack the problem and if there is some reference to this problem such as if it is well known its used, once I read that this have a chaotic behavior, and also read It was used in probability but don't remember where I got that information. So if you know something I would be grateful to be notice, thanks in advance.

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This number is known as the partition number, and there has recently been found a "simple" formula by Ken Ono.

It is not clear to me whether you want $p_k(N)$, which is the number of ways to write $N$ as a sum of $k$ naturals (order not counting), or whether you want $p(N)$, which is the total number of ways of writing $N$ as a sum of natural numbers (order not counting). Of course, $p(N)$ is just the sum of all the values of $p_k(N)$ for $k=1,2,\dots,N$, so the two concepts are closely related, and both have been subject to a lot of study, but, still, the $p(N)$ question is a lot harder than the $p_k(N)$ question - at least for small $k$ there are simple formulas for $p_k(N)$.
Hi @Gerry I was after both as I said I wanted to find $p(N)$ suming $p_{k}(N),k\in {1,2,\cdots,N}$ and also relating $p_{k}(N)$ with $p_{k+1}(N)$ but didn't know that this was a BIG problem. – Ivan3.14 Jul 16 '11 at 21:43