Going back to the ratio test (http://en.wikipedia.org/wiki/Ratio_test), we have:
$$\lim_{n\rightarrow \infty} \sup \frac{a_{p_{n+1}}}{a_{p_{n}}}=R$$
where $R \leq 1$ (it exists because $0 < \frac{a_{p_{n+1}}}{a_{p_{n}}} \leq 1 $, i.e. bounded).
If $R < 1$, then (1)
$$\sum_{n}\frac{a_{n}}{\log n}=\sum_{n}\frac{a_{n}\cdot \pi(n)}{n}\cdot \frac{n}{\pi(n)\cdot \log n}< \left ( 1 + \varepsilon \right )\sum_{n}\frac{a_{n}\cdot \pi(n)}{n}$$
Now, assuming:
$$p_{n}=n_{0}http://en.wikipedia.org/wiki/Bertrand's_postulate)
$$a_{p_{n}}\cdot \pi(p_{n}) \geq \sum_{j=0}^{k}\frac{a_{n_{j}}\cdot \pi(n_{j})}{n_{j}}$$
Or ($\pi(p_{n})=n$):
$$\sum_{n}\frac{a_{n}}{\log n} < \left ( 1 + \varepsilon \right ) \sum_{n}\frac{a_{n}\cdot \pi(n)}{n}\leq \left ( 1 + \varepsilon \right )\sum_{n}n\cdot a_{p_{n}}$$
Now
$$\lim_{n\rightarrow \infty } \sup \frac{(n+1)\cdot a_{p_{n+1}}}{n\cdot a_{p_{n}}}=R< 1$$
It is also worth noting that for $\forall p_{k} \leq n$, $a_{p_{k}} \geq a_{n}$ and as a result:
$$\sum_{k=1}^{\pi (n)}a_{p_{k}}\geq \pi (n)\cdot a_{n}$$
Or:
$$\lim_{n\rightarrow \infty } \pi (n)\cdot a_{n} \leq \sum_{n}a_{p_{n}}< \infty $$
But if we assume
$$\lim_{n\rightarrow \infty } \pi (n)\cdot a_{n}=\gamma >0$$
then from some $n$ we have $\pi (n)\cdot a_{n} > \frac{\gamma }{2}$ or $a_{n} > \frac{\gamma }{\pi (n)\cdot 2}$ or $a_{p_{n}} > \frac{\gamma }{n\cdot 2}$, this will contradict the convergence of the $\sum a_{p}$ and $\gamma =0$. So for $R < 1$ the statement seems to be true.
However for $R=1$, the original condition may not be sufficient.
Let's assume $$\lim_{n\rightarrow \infty} \frac{a_{p_{n+1}}}{a_{p_{n}}}=1$$ strictly. Then, (1) becomes:
$$\left ( 1 - \varepsilon \right )\sum_{n}\frac{a_{n}\cdot \pi(n)}{n} < \sum_{n}\frac{a_{n}}{log(n)}< \left ( 1 + \varepsilon \right )\sum_{n}\frac{a_{n}\cdot \pi(n)}{n}$$
And (2) becomes:
$$\left ( p_{n+1} - p_{n} \right )\cdot \frac{a_{p_{n+1}}\cdot \pi(p_{n})}{p_{n+1}} \leq \sum_{j=0}^{k}\frac{a_{n_{j}}\cdot \pi(n_{j})}{n_{j}} \leq \left ( p_{n+1} - p_{n} \right )\cdot \frac{a_{p_{n}}\cdot \pi(p_{n})}{p_{n}}$$
And:
$$\left ( p_{n+1} - p_{n} \right )\cdot \frac{a_{p_{n+1}}\cdot \pi(p_{n})}{p_{n+1}}=\left ( p_{n+1} - p_{n} \right )\cdot \frac{a_{p_{n}}\cdot \pi(p_{n})}{p_{n}} \cdot \frac{p_{n}}{p_{n+1}} \cdot \frac{a_{p_{n+1}}}{a_{p_{n}}}$$
This indicates (considering the assumption) that:
$$\sum_{n}\frac{a_{n}}{\log n} < \infty \Leftrightarrow \sum_{n} \left ( p_{n+1} - p_{n} \right )\cdot \frac{a_{p_{n}}\cdot \pi(p_{n})}{p_{n}} < \infty \Leftrightarrow \sum_{n} \frac{p_{n+1} - p_{n}}{\log p_{n}} \cdot a_{p_{n}}< \infty$$