Greetings Bobby,
I'd like to run these numbers through Pari again before posting more inconsistent/approximate numbers. The formula with the term "k+1" (see ^{1)} from previous post) is only working properly when CSG = max(0,M)^{2}/gap since M can be negative (because of the aforementioned term). Working out details like these takes me inordinately long...
Good news is, for p = 8,281,634,108,677 and k = 19, I get a CSG > 1 with the roughandready version of the finetuned formula: gap = 1812, M = gap/log(p+gap/2)18 ~ 42.918 (there are 60.918 primes on average in a range of 1812 integers, i.e. 42.918 more than the 18 that are actually between the bounding primes), and CSG = M^{2}/gap ~ 1.0165. With p that large, there won't be much of a difference anymore when using Gram(x) in the calculation of CSG.
