Exponential Power Series where Powers are Prime I am looking for information in regards to a couple particular functions:  
1)  $P(x)=\sum_{p\in\mathbb{P}}\frac{x^p}{p!}$
2)  $Q(x)=\sum_{p\not\in\mathbb{P}}\frac{x^p}{p!}$ (assuming $0, 1$ are included powers in the series...)
3)  $R(x)=\frac{1}{P(x)}$
4)  $S(x)=\frac{1}{Q(x)}$
I don't know if there is much literature on these, since I don't know if they are known, unknown, or what they are called.
 A: trying to relate $P(x) = \sum_{p \in \mathcal{P}} \frac{x^p}{p!}$ to the logarithm of the Riemann zeta function, I get : 
$$P(x) = \sum_{p \in \mathcal{P}} \frac{x^p}{p!} = \frac{x}{(2 i \pi)^2}\int_{|u|=r} \frac{1}{(u-2x)(u-x)}\left( \int_{c-i\infty}^{c+i\infty} (-\ln u)^{-s} \Gamma(s) \sum_{m=1}^\infty \frac{\mu(m)}{m} \ln \zeta(ms) ds\right) du$$
which I hope should be simplifiable.
let :
$$f(z) = \sum_{n=1}^\infty a_n z^n, \qquad\qquad g(z) = \sum_{n=1}^\infty \frac{a_n}{n!} z^n, \qquad \qquad F(s) = \sum_{n=1}^\infty a_n n^{-s}$$ 
(here $a_n = 1$ iff $n$ is prime, hence $F(s) = \sum_{m=1}^\infty \frac{\mu(m)}{m} \ln \zeta(ms)$)
by the Cauchy integral formula, for any $r < R$ the radius of convergence of $f$ : 
$$a_n = \frac{n!}{2 i \pi}\int_{|u|=r} \frac{f(u)}{(u-z)^{n+1}} du$$
hence for any $|z| < r/2$ : $$g(z) = \frac{1}{2 i \pi}\sum_{n=1}^\infty z^n \int_{|u|=r} \frac{f(u)}{(u-z)^{n+1}} du = \frac{1}{2 i \pi}\int_{|u|=r} f(u) \sum_{n=1}^\infty \frac{z^n}{(u-z)^{n+1}} du$$ $$ =\frac{z}{2 i \pi}\int_{|u|=r}  \frac{f(u)}{(u-2z)(u-z)} du$$
while from $n^{-s} \Gamma(s) = \int_0^\infty x^{s-1} e^{-nx} dx$ (extending $f(z)$ on $[0,1[$ by analytic continuation if necessary)  : $$\Gamma(s) F(s) = \int_0^\infty x^{s-1} f(e^{-x})dx$$
hence by inverse Mellin transform : $$f(e^{-x}) = \frac{1}{2 i \pi}\int_{c-i\infty}^{c+i\infty} x^{-s} F(s) \Gamma(s) ds$$
and $$g(z) = \frac{z}{2 i \pi}\int_{|u|=r}  \frac{f(u)}{(u-2z)(u-z)} du = \frac{z}{(2 i \pi)^2}\int_{|u|=r} \frac{1}{(u-2z)(u-z)}\left( \int_{c-i\infty}^{c+i\infty} (-\ln u)^{-s} F(s) \Gamma(s) ds\right) du = \ldots$$
A: From the comment chain on the question, the OP says that some plots would be helpful, so though I cannot answer this question, this is what I can say.
Though finding an analytic expression for these seems a monumental task, approximating them in the first several terms or so is quite easy. I evaluated each of these sums for $0\leq p\leq 41$, and we can estimate their accuracy by evaluating $err(x)=\left|e^x-\sum_{i=1}^{41} x^i/i!\right|$, and noting that if $|x|<4$, $err(x)$ is smaller than $10^{-10}$, so the following graphs can be judged as somewhat accurate.




There are some interesting features. For one, $P(x)$ and $Q(x)$ both have two roots. These are at $0$ and about $-2.301751$ for $P$, and about $-2.24203$ and $-1.05319$ for $Q$. We can also look at a plot of $P$ and $Q$ together with $e^x$:

Note that there are intersections between $P(x)$ and $Q(x)$ at about $2.06337$, $-0.789488$, and $-2.27713$. It is clear in the plot that $P(x)+Q(x)=e^x$.
