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I have two sequences whose relation is described in the following recurrence relations:

$ p_{k + 1} = p_k + \frac{1}{2s_k}$

$ s_{k + 1} = s_k + \frac{s_k}{p_{k+1}}$

(when $p_0=2, s_0 = \frac{1}{2}$ $p_k$ is a probabilistic approximation for the $k+1$th prime number)

I would like to study the asymptotic behavior of the sequences. I have seen the method of generating functions, but this seems to work only if there is one function in the recursive definition. I was able to rewrite some things and get

$ p_{k+1}s_{k+1} = p_ks_k + \frac{1}{2} + s_k $

but this doesn't get me anywhere. Especially, I want to see if $p_k$ approaches $\frac{k}{\log k}$ (like it should do if $p_k$ really was the kth prime number). I was wondering if there is any specific technique or approach to attack this recursive rules.

Ruben
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    You can use the first equation to express $s_k$ interms of $p_k$ and $p_{k+1}$. Then you can substitute into the second equation to get an equation involving only the $p$-sequence. Whether this will do any good, I don't know, but it's worth a try. – Gerry Myerson Aug 10 '14 at 00:24
  • I can't believe I haven't tried that yet. Thanks! – Ruben Aug 10 '14 at 00:35
  • Sadly, it only seems reducible to a non-linear recurrence relation in one variable. – Ruben Aug 10 '14 at 15:20

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