I was playing around with pronic numbers and got a plus and minus reversed. The number I got was not what I expected, but I looked it up on OEIS.org and found:

$\zeta(6)/\zeta(2)/\zeta(3)$

Reducing and inserting my series we get:

$\zeta(3)=\frac{2\pi^4}{315}\prod_{n}^{\infty }\left(1-\frac{1}{p_{n}-p_{n}^{2}}\right)$

Is this a known identity? If not, should I send it somewhere?

Note: the product appears to converge very slowly.