Quote:
Originally Posted by paulunderwood
Results for a=4:
[68368998319, 4]
[416032697359, 4]
[4752200398399, 4]
All semiprimes, p*q, where gcd(p1,q1)=2^i*3^j

No. If g= gcd(p1,q1) it is not g, but (p1)/g and (q1)/g that are 2*i*3^j; in most of the cases g = p1.
Furthermore, in every single case, p1 and q1 are small multiples of a single prime.
I think there's enough numerical evidence to try to devise a script to
construct counterexamples p*q to your test. For example, with a = 4, D = 12, E = 140, find primes l for which (say)
p = 6*l + 1 and q = 36*l + 1 are both prime
p*q is congruent to 1 (mod 560)
kronecker(12,p*q) =  1
[there's another condition I'm too lazy to look up]
The congruence condition mod 560 produces something like 8 possibilities for l (mod 560).
The other examples show the multipliers 6 and 36 aren't the only possibilities.