This may or may not be a surprise, given where I work, but sometimes geeky questions are asked, and geeky responses are necessary. After one such conversation regarding how many ‘a’s are canon for the imperative “Khaaan!”, I decided further investigation was necessary:
It is clear that a strong power law relation is present for low eigenkhans after the initial spike at eigenkhan 1. However, above eigenkhan 100 we enter a new realm that requires further investigation. In both power law regions we have an exponent of ~-2.68 (-2.60 for Google, -2.77 for Bing), which should be considered the canonical khansponent.
This is a slightly extended version as I have more results than I did when I wrote the first email. I can now see that the extended tail above eigenkahn 100 suggests an exponential relation, however I still do not have enough data to create a conclusive model.
In fact, this data follows a distribution known as the Pareto distribution and is related to the Bradford law of diminishing returns. In this case, the distribution has xm = 1 and α = 2.68. The expected value is, then, 1.6 with a variance of 11.12. Therefore, anything from “Kahn!” to “Kaaaaahn!” is within expectations, though there is no incorrect length.
Better things to do with my time? Why yes, why do you ask?