How many primes and semiprimes are generated by this sum? How many primes and semiprimes are generated in the numerator by this sum up to a given $k$? That is, after evaluating the sum and putting the fraction in reduced form, how often is the numerator prime and semiprime?$$ \sum_{n=2}^{k-2}\frac{\pi(n)\pi(k-n)}{\phi(n)\phi(k-n)} $$
I know it fails at $k=15$ when the numerator is $28$.
$\pi(n)$ is the prime counting and $\phi(n)$ is the totient function.
How many primes and semiprimes are generated in the numerator by this sum up to a given $k$?
$$ \sum_{n=2}^k\frac{\pi(n)}{\phi(n)} $$
I know it falis at $k=16$ when the numerator is $40$.
Thanks.
 A: Your first unnamed expression does not have a prime or semiprime numerator for the following values of $k$ in $[3,100]$:
$$   3,  4, 15, 23, 25, 27, 34, 37, 39, 40, 44, 45, 50, 51, 55, 61, 64, 66, 69, 73, 75, 76, 78, 80, 82, 83, 84, 85, 86, 90, 91, 93, 95, 96, 98, 99  \text{.}  $$
It looks like the fraction of $k$ giving prime or semiprime numerators drops below 50% around $k=240$ and continues slowly dropping.  
$k$ giving prime or semiprime numerators up to $k = 400$">
(Testing for primality or semiprimality is taking long enough that I'm unlikely to try to extend this data set.)
Your second unnamed expression does not have a prime or semiprime numerator for the following values of $k$ in $[3,100]$:
$$  16, 25, 29, 34, 38, 54, 55, 56, 57, 59, 62, 63, 64, 67, 70, 73, 74, 75, 79, 80, 82, 83, 85, 86, 87, 88, 90, 92, 93, 96, 98, 100  \text{.}  $$
It seems that this is initially more promising, but drops to 50% also around $k = 240$ and also continues slowly descending.

(Testing for primality or semiprimality is taking long enough that I'm unlikely to try to extend this data set.)
