product $\prod \limits_{i=2}^{\infty} (1+\frac{1}{(p_i -2)p_i})$ for primes $p_i$ I want to calculate the product $\alpha= \prod \limits_{i=2}^{\infty} (1+\frac{1}{(p_i -2)p_i})$ for all primes $p_i >2$.
I calculated this product first with computer and get for the first primes under ten millions $\alpha=1.5147801192603$.
Analytical I tried to expand the product as a telescope sum:
$\alpha=1+\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{2}{45}+\frac{1}{9}-\frac{1}{11}+\frac{2}{105}+\cdots)$. But this doesn't help me. I don't even know if $\alpha$ converges.

Ideas :
The product can be separated as : $\left(\displaystyle \prod \frac{p-1}{p}\right)\left(\displaystyle \prod \frac{p-1}{p-2}\right)$
But we can't separately take both as $\left(\displaystyle \prod \frac{p-1}{p}\right)=\displaystyle \prod (1-p^{-1})=\frac{1}{\zeta(1)}$ is not defined
The other one is harder and concerns the First Hardy-Littlewood conjecture, which is, I think, still not proven.
