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Old 2007-12-16, 18:50   #1
grandpascorpion
 
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Default Expected Number of Factors for numbers within a ra

Hello,

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.
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Old 2007-12-16, 22:33   #2
R. Gerbicz
 
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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.

Last fiddled with by R. Gerbicz on 2007-12-16 at 22:35
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Old 2007-12-17, 13:48   #3
grandpascorpion
 
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Heh, not well enough I guess.

Thanks
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