So this question has strange origins: I was looking at the Leibniz series for $\pi$, and I started to wonder about the relationship between the partial sum and the parity of the value $n$ in the denominator: $2n+1$. (Since these partial sums are related to trig-functions, which ultimately help understand the proof of Euler's formula, I thought this question might be worth thinking about.)
In any case, is there anything particularly interesting about about the value of $n$ in odd-integers? That is, if for some odd number $2n+1$, are the interesting results if $n$ is even or $n$ is odd? (Mersennne primes sort of come to mind, but those are of the form $2^{p}-1$.)
Perhaps as a general direction, can we say something about whether a number is prime, is not prime, is near primes, given we know something about this $n$ (perhaps we could consider cases where $n$ is prime)?
Just wondering. Any thoughts, papers, or further reading would appreciated!