Two brief questions. This seems true but I don't find it using Google. (1) Isn't
$\sum_{k=1}^{\infty} \frac{1}{p(k^2)}$, in which $p(k^2)$ is the $k^2$th prime,
known to converge? I expected to find something in Plouffe around 0.747187, but did not.
It seems that $p(k^2) > k^2$ proves convergence, but then did I miscalculate the constant?
It occurred to me that we might we use comparison (via the PNT) for a series, but all we know is that termwise $\frac{1}{p(k^2)}\sim \frac{1}{k^2\ln k^2},$ and I don't think that $a\sim b$ and $c \sim d$ gives $a+c \sim b+d$ ? So, (2) is it correct that the logarithmic sum also converges but gives no clue as to convergence of the sum in question?