For $\phi(\cdot)$ being Euler's totient function and $n \in \mathbb N_{>1}$, let $S_a=\{n|\phi(n)=a\}$ and $T_a=\{n|n\ne (a+1),2(a+1) \land \phi(n)=a\}$. Plainly $T_a \subset S_a$, and $S_a = T_a \Rightarrow (a+1) \not \in \mathbb P$.
My question is: If we can in principle know $T_a$, can we decide whether $a+1 \in \mathbb P$?
By way of example, $T_{12}=\{21,28,36,42\}$ and $T_{24}=\{35,39,45,52,56,70,72,78,84,90\}$. In these simple cases, we know that $13$ is prime and $25$ is not prime, so $S_{12}=T_{12}+\{13,26\}$, but $S_{24}=T_{24}$. From a knowledge of the members of the sets alone, could we decide that $13$ is prime and $25$ is not?
Assume for some large even number $a$ we did not know whether $a+1$ was prime. In at least some (ideal?) instances, $a$ might be readily factorable such that it would be possible to compute many or all of the members of $T_a$. Under that assumption, is there any mathematical relationship or algorithm that would allow the primality of $a+1$ to be determined from a knowledge of $T_a$? Would a complete knowledge of $T_a$ be necessary, and if not, what fraction of the members of $T_a$ would suffice to determine the primality question?