Unfortunately, no derivative primes were found.

Here are the examples of the tested candidates to give you a flavor of what was done:

Code:

Phi(10,(46421883*2^198800-1)*(299771367*2^198800-1)) is composite: RES64: [DB6C0836F42F050D]
Phi(5,(125526867*2^198800-1)*(299771367*2^198800-1)-1) is composite: RES64: [81248E7866F4AD15]
and a few thousand more...
There are a 200K of similar canidates easily produced and some of them are likely prime and **can** be found
but the cost of sieving and testing is too high to go on for too long.
Besides I already have a couple of primes like these.

What was of primary interest, though, and what I 100% searched were all Phi(5,p), Phi(7,p), Phi(10,p+1), Phi(14,p+1) for all of your several hundred p.

These are more interesting and rare.

They are provable by N-1 because p and p+1 are fully factored which makes

Phi(5,...)-1 and Phi(10,...)-1 50% factored and Phi(7,...)-1 and Phi(14,...)-1 33.333% factored which is enough got the proof.

Other polynomials can be tried (and tried and tried until a constructible prime is found) but they are not as interesting as Phi().

I have

one of these but it was not eligible for Top5000. Just a little too small.