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Functions that count the number of numbers containing k factors (with multiplicity, k = 1,2,3,...) on an interval arise in connection with generalizations of the P.N.T. I wonder if someone can provide a reference to work involving the distribution of distances between them? For example (calling these numbers k-primes)

$F_{n,k}(d) :=$ card {k-primes on $[2^n,2^{(n+2)}]$ with $( p_{k,m+1} - p_{k,m} ) = d \}$,

in which $p_{k,m} $ is the $m$th k-prime in the interval.

For many such intervals it appears that for given k the distribution is nearly optimal in the sense of minimizing the variance of the distances. At any rate I imagine it's well-studied.

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There might be something in D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, Small gaps between primes or almost primes, http://www.arxiv.org/abs/math.NT/0506067/.

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Thank you. I didn't see this until today. –  daniel Nov 26 '11 at 17:45
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