Set and subsets link by prime numbers I have a bit idea to solve this problem for small $n$ by programation but I think for $n>100$ I will need maths to help me. My problem is :

Let S be the set of prime numbers less than n.
  Find the number of subsets of S, the sum of whose elements is a prime number.

This problem seems to be very difficult, I'm woriking on it for few month when I have time, but for the moment I saw no possible symetries in the problem. It's "not difficult" to find all subsets the sum of whose elements is an even number but it just can give me an approximation of the number of all subsets and not the exact number. 
Any hints or partial answers will be appreciated 
PS : I'm a chemist maths is not my work, however if you have a very technical thing to say, say it please, if I have to take 4 months to understand I will take them, I like learn and have an intensive brain activity ! 
 A: Well let $p_k$ be the $k$-th prime and $f(k,n)$ be the number of subsets of the first $k$ primes that have a sum of $n$. Then obviously $f(k,n) = f(k-1,n) + f(k-1,n-p_k)$ for any $k \in \mathbb{N}^+$ and $n \in [1..\sum_{i=1}^{k-1} p_i]$. Since $\sum_{i=1}^{k-1} p_i \in O(k^2 \ln(k))$ as $k \to \infty$ by the prime number theorem, this algorithm will take $O(k^3\ln(k))$ time for $k$ primes, which is $O\Big( \frac{n^3}{\ln(n)^2} \Big)$ for primes below $n$. After this we can just find the sum of $f(k,q)$ over all primes $q \in [1..n^2]$ to get the desired number for the first $k$ primes below $n$, which takes $O(n^2)$ using the sieve method. I doubt there is any better approach to this problem because of the (lack of) structure.
Edit
As Chris Culter below pointed out, if we want the exact answer rather than to a certain fixed precision, then the time complexity becomes much higher because the answer grows exponentially with respect to $n$. I'm not sure what is the exact asymptotic behaviour, but given any $k$, heuristically taking $n \sim \frac{1}{2} \sum_{i=1}^k p_i \sim \frac{1}{4} {p_k}^2 ( \ln(p_k) - 1 )$ should roughly maximize $f(k,n)$ with value roughly about $2^{O(k)}$, and hence the computations of $f(k,n)$ for a large number of values of $n$ would take about $O(k)$ time each. This would give a total of roughly $O( k^3 \ln(k) )$ time for computing all the values of $f(k,n)$ for a given $k$, and hence $O( k^4 \ln(k) )$ time in total for $k$ primes, which is $O\Big( \frac{n^4}{\ln(n)^3} \Big)$ time for primes below $n$. All this would not be relevant if we only want to compute the answer to a fixed number of significant digits.
A: As a reference point, there are 17193362135842433291 prime sums among primes up to $331$ (the 67th prime). That's as far as we can go using 64-bit arithmetic. For more values, see http://oeis.org/A071810 "Number of subsets of the first $n$ primes whose sum is a prime", where T. D. Noe has contributed a list up to the 100th prime.
