For the number of partitions of n into prime parts $a(n)$ it holds
$$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$
where $q(n)$ the sum of all different prime factors of $n$.
Due to https://oeis.org/A000607.
I have found that $$a(p_n)\approx a(p_{n-2})+a(p_{n-1})\tag 2$$ and conjecture the asymptotic relation.
$$\log a(p_n)\sim \log \big(a(p_{n-2})+a(p_{n-1})\big )\tag 3$$
On x-axis the prime numbers $p_n$ are plotted. The blue lines correspond to $a(p_n)$ and the red lines correspond to $a(p_{n-2})+a(p_{n-1})$.
On the oeis site above there is also a formula $$a(n)\sim e^{2\pi\sqrt{n/\log n}\,/\sqrt{3}}\tag 4$$ but I don't know if this really helps?
Can this (reformulated) conjecture be proved?