2007-12-16, 18:50 | #1 |
Jan 2005
Transdniestr
503 Posts |
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. |
2007-12-16, 22:33 | #2 |
"Robert Gerbicz"
Oct 2005
Hungary
2×3×11×23 Posts |
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 |
2007-12-17, 13:48 | #3 |
Jan 2005
Transdniestr
503 Posts |
Heh, not well enough I guess.
Thanks |
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