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Old 2014-08-02, 21:48   #3
R.D. Silverman
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"Bob Silverman"
Nov 2003
North of Boston

1D2416 Posts

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).

\rho_0(n) is the standard Dickman-de Bruijn function.

The higher k's can (I think) be defined easily in terms of an integral of (k-1) functions.

I'd hope that this heuristic gives a probability density for the number of large primes, with "large" defined by the choice of n. 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.
Wrong approach. What needs to be computed is the conditional
probability that a large integer N has k prime factors given that
it has no factors less than (say) N^1/a, for given a.
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