In an earlier post, I presented general formulae for the number of “candidate” k-tuples within the n-th primorial after n rounds of sieving by successive primes to the n-th prime, Pn, as follows:
Where Kn is a generalized number of candidate k-tuples within the primorial Pn#, Tn is the number of candidate twin pairs, Rn the number of candidate triplets, Qn quadruplets, and QTn quintuplets.
I’ve previously stated that we should expect the actual number of primes or k-tuples less than Pn^2 to be about equal to the density of candidate k-tuples within the n-th primorial times Pn^2. In other words, we present the estimate:
pi(Pn^2) ~ (Kn/Pn#)*Pn^2
A couple of posts back, I compared the results of this prediction for primes and twin primes with their actual counts up to some fairly large number (Pn ~ 1150, Pn^2 ~ 1.3 x 10^6). In this post I simply want to add similar comparisons for the prime triplets, quadruplets and quintuplets in the following two graphs:
Again, we see reasonably good correspondence, but a slight over-prediction that increases with n and Pn.