What is the number of inter partitions of an arbitrary positive integer $n$ that consist of only primes? I would like to know that when given a positive integer, how to find the number of sets of only prime numbers that sum up to it.  For example, if $n=13$, one such set comes from $3 + 5 + 5 = 13$.
Is there a formula for this? I am trying for the number $13$, there is $9$ in total but I can only find $8$.

A professional mathematician would ask the problem this way:
What is the number of integer partitions of an arbitrary positive integer $n$ that consist of only primes?
 A: According to A000607 there dosen't seem to be an easy formula.
In this sequence it counts partitions, the number itself as a sum is a partition.
A: Here's some context. In general, the generating function for the number $p_S(n)$ ways to partition an integer $n$ into a sum of elements of a subset $S \subseteq \mathbb{N}$ of the natural numbers is
$$f_S(z) = \prod_{s \in S} \frac{1}{1 - z^s}.$$
If $S$ is finite these are rational functions and closed forms for $p_S(n)$ can be given; for example, if $S = \{ 1, 5, 10, 25 \}$ we are counting the number of ways to make change for $n$ cents using pennies, nickels, dimes, and quarters.
These are very complicated functions if $S$ is infinite. When $S = \mathbb{N}$ we get what is just called the partition function $p_{\mathbb{N}}(n) = p(n)$, with generating function
$$f_{\mathbb{N}}(z) = \frac{1}{(1 - z)(1 - z^2)(1 - z^3) \dots}.$$
The infinite product appearing in the denominator is the Euler function. $p(n)$ has no closed form but Hardy and Ramanujan famously showed that it is asymptotic to
$$p(n) \sim \frac{1}{4n \sqrt{3}} \exp \left( \pi \sqrt{ \frac{2n}{3} } \right)$$
and a complete asymptotic expansion is known and due to Rademacher. I say all this to emphasize that even when $S = \mathbb{N}$ this is a complex and deep problem.
Nevertheless some things can be said. In the context of Flajolet and Sedgewick's Analytic Combinatorics problems of this type can be dealt with using saddle-point asymptotics, which are at least good enough to give asymptotics for the logarithm
$$\boxed{ \log p_{\mathbb{P}}(n) \sim 2\pi \sqrt{ \frac{n}{3 \log n} } }$$
where $\mathbb{P}$ is the primes; this is VIII.26. I don't know the details of the analysis unfortunately.
A: I don't know a formula, but it is very simple coding:  In Mathematica:
n=13;
Select[IntegerPartitions[n], AllTrue[#, PrimeQ] &]

which gives the nine sequences:
(*
{{13}, {11, 2}, {7, 3, 3}, {7, 2, 2, 2}, {5, 5, 3}, {5, 3, 3, 2},
{5, 2, 2, 2, 2}, {3, 3, 3, 2, 2}, {3, 2, 2, 2, 2, 2}}
*)
for $n=8$:
(* {{5, 3}, {3, 3, 2}, {2, 2, 2, 2}} *)
The number of such sequences (comprising only primes) increases very rapidly with $n$:

