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Old 2014-08-03, 05:42   #5
CRGreathouse's Avatar
Aug 2006

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Originally Posted by wblipp View Post
How about building on the Dickman-de Bruijn function by defining a series

\rho_k(n) is the asymptotic probability that x has exactly k prime divisors greater than x^(1/n).
Hmm. I think I could do this, but it would be a real bear to compute. I wrote code to compute, or at least closely estimate, that function a few years back and it wasn't easy. It seems like this version would suffer from all the problems of the original, but worse.

But at the ranges that I've tested it the Dickman function already does a fairly poor job; not sure how these would fare, better or worse.

Originally Posted by wblipp View Post
You could use the knowledge of prior factoring effort to choose interesting definitions of "large," and apply Bayes' theorem to modify the density to account for the prior effort.
Sounds reasonable.
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