In this previous question there was shown that: Let $a_{n,k}$ denote the number of ordered ways of writing $n$ as a sum of $k$ primes. Then after some calculation, one finds that: $$a_{n,k} = \frac{k}{n-2k} \sum_{v=0}^{n-1} {a_{v,k} b_{n-1-v}}$$ which is a recurence relation. I am looking at $k=2...

Let $I(n)$ be the set of the first $n$ odd integers, $S_p(n)$ be a subset of the last $p$ elements of $I(n)$ such that $p$ is some prime number, and $a_{n-\left(\frac{p-1}{2}\right)}$ the $n-\left(\frac{p-1}{2}\right)_{th}$ element of $I(n)$. Thinking on the Goldbach conjecture, I came to the fol...