Browsing through OESIS I have found that if $p_p(n)$ denotes the number of partitions of $n$ into prime parts then $p_p(n) = O(e^{\frac{2 \Pi}{\sqrt{3}}\sqrt{n/\log n}})$.

I am interested in the asymptotic behaviour of a more specific function - $p(n,k)$ defined as the number of partitions of $n$ into $k$ parts such that every part is an odd prime. (for example one such partition of 13 would be 7+3+3)

Is there any known literature for looking up such identities? Or perhaps, is there an easy way to derive an asymptotic bound for $p(n,k)$ ?