How do we calculate
the weight for a form (k*b^n+c)/gcd(k+c,b-1) for fixed integers k >= 1, b >= 2, c != 0, gcd(k,c) = 1, gcd(b,c) = 1.
I have a
research that whether the form (k*b^n+c)/gcd(k+c,b-1) can contain a prime (there are special cases, such as (27*8^n+1)/7, (1*8^n-1)/7, (1*16^n-1)/15, and (4*16^n+1)/5, they can contain
only one prime (because of the algebra factors), thus the weight is also 0, besides, there are also cases without covering set or algebra factors, but cannot contain a prime, such as 8*128^n+1 and 32*128^n+1)
(I know that the
dual forms have the same weight)