Asymptotics for the number of partitions of $n$ into odd prime parts Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any paper or book studying the mentioned quantity but the amount of literature available to me is quite limited and I am wondering if someone could tell me what the asymptotic behavior of $p_o(n)$ is and perhaps point out a relevant reference for me to read.
Thanks!
 A: Flajolet and Sedgewick, Analytic Combinatorics (link goes to free, legal downloadable PDF of book), section VIII.6 treats the asymptotics of various types of partitions. They get that the number of partitions of $n$ into prime parts, which they denote $P_n^{(\Pi)}$ (and which is A000607) satisfies $\log P_n^{(\Pi)} \sim (2\pi/\sqrt{3}) \sqrt{n/\log n}$ (Here $f(n) \sim g(n)$ has the usual meaning $\lim_{n \to \infty} f(n)/g(n) = 1$.) 
I believe (but have not explicitly checked) that if you disallow any finite number of primes this asymptotic formula still holds; in particular it should hold in the case you're asking about.
Edited to add: The logarithmic growth rate comes from a saddle-point estimate which can in turn be derived from the rate of growth of $\prod_{n \ge 1} 1/(1-z^{p_n})$ (where $p_n$ is the $n$th prime) as $z \to 1^-$ along the real axis. As stated after the equation (73) in F&S, p. 580, 
$$ \sum_{n \ge 1} e^{-tp_n} \sim {t \over \log t} $$
and it's from this that the asymptotic result is derived. (Note that $z \to 1^-$ corresponds to $t \to 0^+$.) But if we omit the single term $e^{-2t}$ from that sum it won't change the asymptotic behavior at 0.
This isn't a full proof, though. The result is also a little bit counterintuitive, since one expects that a random partition of $n$ into primes, for large $n$, will contain at least one 2, and therefore leaving the 2s out should reduce the count considerably. But on the logarithmic scale this appears to be negligible.
A: I think I found the answer I was looking for in some old paper by Erdos. http://www.renyi.hu/~p_erdos/1942-02.pdf
On page 448 he says:
Let $a_1 < a_2 < ...$ be an infinite sequence of integers of density $\alpha$ such that the a's have no common factor. Denote by $p'(n)$ the number of partitions of $n$ into the a's. Then $$\log(p'(n)) = c(\alpha n)^{1/2}.$$ where $c = \pi \sqrt{\frac{2}{3}}$
I just have to make sure what precisely he meant with the term density as in the classical sense http://en.wikipedia.org/wiki/Asymptotic_density density of a sequence is defined as a number and in this case the asymptotic identity he derived makes no sense. If someone knows what is the precise definition of density in this case, let me know it as a comment please!
