The interesting q and p are when they produce a special large factor:
If a characteristic factor q, in our list is also the largest factor
yet, it divides a Lucas number of the form: L(2^n * p^m)
p is a prime, n=>0, m=>0, Limits for n,m?
Here is a list of Lucas indicies, in a progession due to their
increasing factor size.
These indicies, without the powers of two.
I would be highly interested in the density of these primes!
Fibonacci's have a similar probable form.