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!



Asymptotics for the number of partitions of $n$ into odd prime parts
Flajolet and Sedgewick, [Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf) (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](http://oeis.org/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.


