Let $P(n)$ is number of partitions of $n$ into natural numbers. $R(n)$ is number of partitions of $n$ into prime numbers.
Is there any expression that relates $P(n)$ , and $R(n)$? I look for something that looks like this: $R(n) = f(P(n))$, for some function $f$.
Here is what have i done.
Consider two generating functions:
$$G(x) := \prod_{n = 1}^{\infty}\frac{1}{1-x^n} = \sum_{m=0}^{\infty}P(n)x^n$$ $$F(x): = \prod_{p\in\mathbb{P}}\frac{1}{1-x^p}=\sum_{m=0}^{\infty}R(n)x^n$$.
I was trying to apply inclusion-exclusion formula to find connection between $G$ and $F$. For example $$\frac{(1-x)G(x)}{F(x)}=\prod_{n\in \mathbb{N}\setminus(\{1\}\cup \mathbb{P})}\frac{1}{1-x^n}$$ And this is related to the number of partitions into composite numbers.
Thank you in advance for any help.