What is a good sieve program for

Generalized Pierpoint primes which do not have the form k*b^n+-1 where (k < 2^32) is small? That is, a sieve which works on primes of the form 2^a*p^b*q^c*r^d...(r_n)^(d_n) where p, q, r..., r_n are distinct odd primes and there are NO restrictions on the exponents a, b, c, d,... d_n, such as their size. Does the sieve work when trying to find primes of the form 2^a*p^b*q^c*r^d...(r_n)^(d_n) where all the primes (p, q, r,..., r_n) are fixed, and all the exponents for the primes are fixed (except only one, two, or even three primes).

For example, I found a prime of the form 2^a*3^b*5^c*7^d+1 with no definite ratio, pattern, or restrictions for the exponents a, b, c, d. Here is what my sieve file looked like:

(I chose the fixed exponents for 2 and 3 randomly and the exponent ranges for 5 and 7 randomly)

sieve.txt:

---

ABC2 2^1473*3^2731*5^$a*7^$b+1

a: from 1000 to 1000

b: from 400 to 500

---

Running the program up to b = 415, I only found one PRP (which was later proved prime):

2^1473*3^2731*5^1020*7^408+1

---

If I wanted to try higher fixed exponents and ranges, what would be a good sieve program to use so I know which numbers I should test? Thanks for help.