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grandpascorpion 2007-12-16 18:50

Expected Number of Factors for numbers within a ra

I was wondering if there's a simple formula to estimate the number of prime factors for a number in some range say a to b.

For instance, between x=900 and x=1000, the expected number of prime factors is 3.04 ~ 3.

R. Gerbicz 2007-12-16 22:33

It's very well known that the expected number of different prime factors for a large positive integer N is log(log(N))+O(1), this is also true if you write multiplicity instead of different. So the answer is also log(log(b))+O(1) for your question, supposing that b is large and a isn't very close to b.

You can prove this using Merten's theorem: sumprime(p=2,n,1/p)=log(log(n))+O(1) is enough for you.

grandpascorpion 2007-12-17 13:48

Heh, not well enough I guess.


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