Actual versus predicted k-tuples less than Pn^2

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.


About petequinn

I'm a Canadian geotechnical engineer specializing in the study of landslides. I started this page to discuss some mathematical topics that interest me, initially this involved mostly prime numbers, but more recently I've diverted focus back to a number of topics of interest in geotechnique, geographic information systems and risk. I completed undergraduate training in engineering physics at Royal Military College (Kingston, Ontario), did a masters degree in civil (geotechnical) engineering at University of British Columbia (Vancouver), and doctorate in geological engineering at Queen's University (Kingston). I was a military engineer for several years at the beginning of my career, and did design and construction work across Canada and abroad. I've worked a few years for the federal government managing large environmental clean up projects in Canada's arctic, and I've worked across Canada, on both coasts and in the middle, as a consulting geotechnical engineer. My work has taken me everywhere in Canada's north, to most major Canadian cities and many small Canadian towns, and to Alaska, Chile, Bermuda, the Caribbean, Germany, Norway, Sweden, Bosnia, and Croatia. My main "hobby" is competitive distance running, which I may write about in future.
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