I found some unexpected data for primes. Consider $p(n)$ being the product over all primes smaller than or equal to $n$. When factoring $p(n)^a +1$ for $a=1$ or $a=2$ we get the expected amount of primes.
For instance for $a=2$ , $p(7)^2+1$ is prime and we get primes for $p(T(n))^2+1$ where $T(n)=e^{2n}+O(\sqrt {ln(n)})$ as expected.
However for $a=3$ we expect primes for $p(T_3(n))^2+1$ where $T_3(n)=e^{3n}+O(\sqrt {ln(n)})$.
However for $1<n<61$ , $p(n)^3+1$ is never a prime. And up to $n=101$ I find at best semiprimes and some pseudoprimes.
This contradicts statistics based on the prime number theorem alot !!
I havent seen such an unexpected result since the MERSENNE VS FERMAT primes.
How to explain this ?
Recall that $x^3+1=(x+1)(x^2-x+1)$.
The same non-primes phenomenon will occur whenever $a$ has an odd divisor greater than $1$, that is, whenever $a$ is not a power of $2$.