Define $S_{odd}$ as all $n\in N $ where $n$ is the product of an odd number of distinct primes. Define $S_{even}$ similarly. Thus: $$S_{odd} = \{2,3,5,...,30,42,....\}$$
$$S_{even} = \{6,10,14, ....,210,...\}$$
To be clear, neither contains a multiple of a square
Then: $$\sum_{s\in S_{odd}} 1/s^2 - \sum_{s\in S_{even}} 1/s^2 = 1 - 6/\pi^2$$ I think we can maybe write the left hand-side in terms of $\sum_{p\in P} 1/p^k$, but it's a bit of a puzzle. What are you thoughts?